4cycl4dv4e - Metamath Proof Explorer

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Theorem 4cycl4dv4e 26196

Description: If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)

**Assertion**
Ref	Expression
4cycl4dv4e	⊢ ((𝑉 USGrph 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))

Distinct variable groups: 𝐸,𝑎,𝑏,𝑐,𝑑 𝑃,𝑎,𝑏,𝑐,𝑑 𝑉,𝑎,𝑏,𝑐,𝑑 Allowed substitution hints: 𝐹(𝑎,𝑏,𝑐,𝑑)

Proof of Theorem 4cycl4dv4e Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycliswlk 26160	. . . . 5 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → 𝐹(𝑉 Walks 𝐸)𝑃)
2		wlkbprop 26051	. . . . 5 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
3	1, 2	syl 17	. . . 4 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
4		iscycl 26153	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
5		ispth 26098	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
6		istrl 26067	. . . . . . . . . 10 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
7		fzo0to42pr 12422	. . . . . . . . . . . . . . . . . . 19 ⊢ (0..^4) = ({0, 1} ∪ {2, 3})
8	7	raleqi 3119	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ ({0, 1} ∪ {2, 3})(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})
9		ralunb 3756	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑘 ∈ ({0, 1} ∪ {2, 3})(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ ∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
10		2wlklem 26094	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
11		2z 11286	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℤ
12		3z 11287	. . . . . . . . . . . . . . . . . . . 20 ⊢ 3 ∈ ℤ
13		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 2 → (𝐹‘𝑘) = (𝐹‘2))
14	13	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 2 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘2)))
15		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
16		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
17		2p1e3 11028	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 + 1) = 3
18	16, 17	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 2 → (𝑘 + 1) = 3)
19	18	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
20	15, 19	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 2 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
21	14, 20	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 2 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
22		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 3 → (𝐹‘𝑘) = (𝐹‘3))
23	22	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 3 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘3)))
24		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 3 → (𝑃‘𝑘) = (𝑃‘3))
25		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 3 → (𝑘 + 1) = (3 + 1))
26		3p1e4 11030	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (3 + 1) = 4
27	25, 26	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 3 → (𝑘 + 1) = 4)
28	27	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 3 → (𝑃‘(𝑘 + 1)) = (𝑃‘4))
29	24, 28	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 3 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘3), (𝑃‘4)})
30	23, 29	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 3 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}))
31	21, 30	ralprg 4181	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})))
32	11, 12, 31	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}))
33	10, 32	anbi12i 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ ∀𝑘 ∈ {2, 3} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})))
34	8, 9, 33	3bitri 285	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})))
35		preq2 4213	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑃‘4) = (𝑃‘0) → {(𝑃‘3), (𝑃‘4)} = {(𝑃‘3), (𝑃‘0)})
36	35	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃‘0) = (𝑃‘4) → {(𝑃‘3), (𝑃‘4)} = {(𝑃‘3), (𝑃‘0)})
37	36	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘4) → ((𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)} ↔ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))
38	37	anbi2d 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘0) = (𝑃‘4) → (((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)}) ↔ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})))
39	38	anbi2d 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃‘0) = (𝑃‘4) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) ↔ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))))
40		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) → 𝑉 USGrph 𝐸)
41		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) → 𝐹 ∈ Word dom 𝐸)
42		simplrr 797	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) → Fun ^◡𝐹)
43		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) → (#‘𝐹) = 4)
44		4cycl4dv 26195	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹 ∧ (#‘𝐹) = 4)) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3))))))
45	40, 41, 42, 43, 44	syl13anc 1320	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3))))))
46	45	imp 444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) ∧ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))) → ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)))))
47		4z 11288	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 4 ∈ ℤ
48		3re 10971	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 3 ∈ ℝ
49		4re 10974	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 4 ∈ ℝ
50		3lt4 11074	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 3 < 4
51	48, 49, 50	ltleii 10039	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 3 ≤ 4
52		eluz2 11569	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (4 ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ 4 ∈ ℤ ∧ 3 ≤ 4))
53	12, 47, 51, 52	mpbir3an 1237	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 4 ∈ (ℤ_≥‘3)
54		4fvwrd4 12328	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((4 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...4)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
55	53, 54	mpan 702	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
56		preq12 4214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
57	56	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
58	57	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸))
59		simplr 788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘1) = 𝑏)
60		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘2) = 𝑐)
61	59, 60	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘1), (𝑃‘2)} = {𝑏, 𝑐})
62	61	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ↔ {𝑏, 𝑐} ∈ ran 𝐸))
63	58, 62	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ↔ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
64		preq12 4214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) → {(𝑃‘2), (𝑃‘3)} = {𝑐, 𝑑})
65	64	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘2), (𝑃‘3)} = {𝑐, 𝑑})
66	65	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ↔ {𝑐, 𝑑} ∈ ran 𝐸))
67		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘3) = 𝑑)
68		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (𝑃‘0) = 𝑎)
69	67, 68	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → {(𝑃‘3), (𝑃‘0)} = {𝑑, 𝑎})
70	69	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ({(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸 ↔ {𝑑, 𝑎} ∈ ran 𝐸))
71	66, 70	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸) ↔ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)))
72	63, 71	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ↔ (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸))))
73	68, 59	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘0) ≠ (𝑃‘1) ↔ 𝑎 ≠ 𝑏))
74	68, 60	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘0) ≠ (𝑃‘2) ↔ 𝑎 ≠ 𝑐))
75	68, 67	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘0) ≠ (𝑃‘3) ↔ 𝑎 ≠ 𝑑))
76	73, 74, 75	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ↔ (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑)))
77	59, 60	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘1) ≠ (𝑃‘2) ↔ 𝑏 ≠ 𝑐))
78	59, 67	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘1) ≠ (𝑃‘3) ↔ 𝑏 ≠ 𝑑))
79		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) → (𝑃‘2) = 𝑐)
80		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) → (𝑃‘3) = 𝑑)
81	79, 80	neeq12d 2843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) → ((𝑃‘2) ≠ (𝑃‘3) ↔ 𝑐 ≠ 𝑑))
82	81	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘2) ≠ (𝑃‘3) ↔ 𝑐 ≠ 𝑑))
83	77, 78, 82	3anbi123d 1391	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)) ↔ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))
84	76, 83	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3))) ↔ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))
85	72, 84	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)))) ↔ ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))
86	85	biimpcd 238	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)))) → ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))
87	86	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)))) → (∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))
88	87	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)))) → (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))
89	88	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)))) → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))
90	89	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸) ∧ ({(𝑃‘2), (𝑃‘3)} ∈ ran 𝐸 ∧ {(𝑃‘3), (𝑃‘0)} ∈ ran 𝐸)) ∧ (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘0) ≠ (𝑃‘3)) ∧ ((𝑃‘1) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘3) ∧ (𝑃‘2) ≠ (𝑃‘3)))) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))
91	46, 55, 90	syl2im 39	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑉 USGrph 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹)) ∧ (#‘𝐹) = 4) ∧ (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)}))) → (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))
92	91	exp41 636	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑉 USGrph 𝐸 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((#‘𝐹) = 4 → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))))
93	92	com14 94	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((#‘𝐹) = 4 → (𝑉 USGrph 𝐸 → (𝑃:(0...4)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))))
94	93	com35 96	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘0)})) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))))
95	39, 94	syl6bi 242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃‘0) = (𝑃‘4) → ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))))
96	95	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) → ((𝑃‘0) = (𝑃‘4) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))))
97	96	com24 93	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}) ∧ ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ∧ (𝐸‘(𝐹‘3)) = {(𝑃‘3), (𝑃‘4)})) → (𝑃:(0...4)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((𝑃‘0) = (𝑃‘4) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))))
98	34, 97	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝑃:(0...4)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((𝑃‘0) = (𝑃‘4) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))))
99	98	com13 86	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...4)⟶𝑉 → (∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → ((𝑃‘0) = (𝑃‘4) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))))
100	99	3imp 1249	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘4) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
101	100	com14 94	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘4) → (𝑉 USGrph 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
102		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 4 → (𝑃‘(#‘𝐹)) = (𝑃‘4))
103	102	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘4)))
104		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 4 → (0...(#‘𝐹)) = (0...4))
105	104	feq2d 5944	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 4 → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...4)⟶𝑉))
106		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 4 → (0..^(#‘𝐹)) = (0..^4))
107	106	raleqdv 3121	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 4 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
108	105, 107	3anbi23d 1394	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 4 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
109	108	imbi1d 330	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 4 → ((((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))) ↔ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))
110	109	imbi2d 329	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 4 → ((𝑉 USGrph 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))) ↔ (𝑉 USGrph 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...4)⟶𝑉 ∧ ∀𝑘 ∈ (0..^4)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
111	101, 103, 110	3imtr4d 282	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 4 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
112	111	com14 94	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
113	112	2a1d 26	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))))
114	6, 113	syl6bi 242	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))))))
115	114	3impd 1273	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))))
116	5, 115	sylbid 229	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))))
117	116	impd 446	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
118	4, 117	sylbid 229	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
119	118	3adant1 1072	. . . 4 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))))))
120	3, 119	mpcom 37	. . 3 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → (𝑉 USGrph 𝐸 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))
121	120	com12 32	. 2 ⊢ (𝑉 USGrph 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 4 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))))
122	121	3imp 1249	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ ({𝑐, 𝑑} ∈ ran 𝐸 ∧ {𝑑, 𝑎} ∈ ran 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {cpr 4127 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 2c2 10947 3c3 10948 4c4 10949 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 USGrph cusg 25859 Walks cwalk 26026 Trails ctrail 26027 Paths cpath 26028 Cycles ccycl 26035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-usgra 25862 df-wlk 26036 df-trail 26037 df-pth 26038 df-cycl 26041

This theorem is referenced by: n4cyclfrgra 26545

W3C validator