Theorem sosng 4413

Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Assertion

Ref	Expression
sosng	⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem sosng

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sopo 4050	. . 3 ⊢ (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2		posng 4412	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
3	1, 2	syl5ib 143	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} → ¬ 𝐴𝑅𝐴))
4	2	biimpar 281	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Po {𝐴})
5		ax-in2 545	. . . . . . . . 9 ⊢ (¬ 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
6	5	adantr 261	. . . . . . . 8 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
7		elsni 3393	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8		elsni 3393	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
9	7, 8	breqan12d 3779	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐴))
10	9	imbi1d 220	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
11	10	adantl 262	. . . . . . . 8 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
12	6, 11	mpbird 156	. . . . . . 7 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
13	12	ralrimivw 2393	. . . . . 6 ⊢ ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
14	13	ralrimivva 2401	. . . . 5 ⊢ (¬ 𝐴𝑅𝐴 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
15	14	adantl 262	. . . 4 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
16		df-iso 4034	. . . 4 ⊢ (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))
17	4, 15, 16	sylanbrc 394	. . 3 ⊢ (((Rel 𝑅 ∧ 𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Or {𝐴})
18	17	ex 108	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (¬ 𝐴𝑅𝐴 → 𝑅 Or {𝐴}))
19	3, 18	impbid 120	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∈ wcel 1393  ∀wral 2306  Vcvv 2557  {csn 3375   class class class wbr 3764   Po wpo 4031   Or wor 4032  Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034

This theorem is referenced by: (None)