Theorem nnaordex 6100

Description: Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnaordex	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnaordex

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

Step	Hyp	Ref	Expression
1		eleq2 2101	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2049	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = 𝐵))
3	2	anbi2d 437	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
4	3	rexbidv 2327	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
5	1, 4	imbi12d 223	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
6	5	imbi2d 219	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ ω → (𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏))) ↔ (𝐴 ∈ ω → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))))
7		eleq2 2101	. . . . . 6 ⊢ (𝑏 = ∅ → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ ∅))
8		eqeq2 2049	. . . . . . . 8 ⊢ (𝑏 = ∅ → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = ∅))
9	8	anbi2d 437	. . . . . . 7 ⊢ (𝑏 = ∅ → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅)))
10	9	rexbidv 2327	. . . . . 6 ⊢ (𝑏 = ∅ → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅)))
11	7, 10	imbi12d 223	. . . . 5 ⊢ (𝑏 = ∅ → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅))))
12		eleq2 2101	. . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝑦))
13		eqeq2 2049	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = 𝑦))
14	13	anbi2d 437	. . . . . . 7 ⊢ (𝑏 = 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)))
15	14	rexbidv 2327	. . . . . 6 ⊢ (𝑏 = 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)))
16	12, 15	imbi12d 223	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))))
17		eleq2 2101	. . . . . 6 ⊢ (𝑏 = suc 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ suc 𝑦))
18		eqeq2 2049	. . . . . . . 8 ⊢ (𝑏 = suc 𝑦 → ((𝐴 +𝑜 𝑥) = 𝑏 ↔ (𝐴 +𝑜 𝑥) = suc 𝑦))
19	18	anbi2d 437	. . . . . . 7 ⊢ (𝑏 = suc 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
20	19	rexbidv 2327	. . . . . 6 ⊢ (𝑏 = suc 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
21	17, 20	imbi12d 223	. . . . 5 ⊢ (𝑏 = suc 𝑦 → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏)) ↔ (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))))
22		noel 3228	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
23	22	pm2.21i 575	. . . . . 6 ⊢ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅))
24	23	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = ∅)))
25		elsuci 4140	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
26		simpr 103	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)))
27		peano2 4318	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
28	27	ad2antlr 458	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → suc 𝑥 ∈ ω)
29		elelsuc 4146	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥)
30	29	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥))
31		nnasuc 6055	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
32		suceq 4139	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 +𝑜 𝑥) = 𝑦 → suc (𝐴 +𝑜 𝑥) = suc 𝑦)
33	31, 32	sylan9eq 2092	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (𝐴 +𝑜 suc 𝑥) = suc 𝑦)
34	33	ex 108	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝑦 → (𝐴 +𝑜 suc 𝑥) = suc 𝑦))
35	30, 34	anim12d 318	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦)))
36	35	imp 115	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦))
37		eleq2 2101	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ suc 𝑥))
38		oveq2 5520	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑥 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 suc 𝑥))
39	38	eqeq1d 2048	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → ((𝐴 +𝑜 𝑧) = suc 𝑦 ↔ (𝐴 +𝑜 suc 𝑥) = suc 𝑦))
40	37, 39	anbi12d 442	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦)))
41	40	rspcev 2656	. . . . . . . . . . . . . 14 ⊢ ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +𝑜 suc 𝑥) = suc 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦))
42	28, 36, 41	syl2anc 391	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦))
43	42	ex 108	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦)))
44	43	rexlimdva 2433	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦)))
45		eleq2 2101	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑥))
46		oveq2 5520	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑥))
47	46	eqeq1d 2048	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝐴 +𝑜 𝑧) = suc 𝑦 ↔ (𝐴 +𝑜 𝑥) = suc 𝑦))
48	45, 47	anbi12d 442	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
49	48	cbvrexv 2534	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +𝑜 𝑧) = suc 𝑦) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
50	44, 49	syl6ib 150	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
51	50	ad2antlr 458	. . . . . . . . 9 ⊢ (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
52	26, 51	syld 40	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
53		0lt1o 6023	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1𝑜
54	53	a1i 9	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∅ ∈ 1𝑜)
55		nnon 4332	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
56		oa1suc 6047	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 1𝑜) = suc 𝐴)
57	55, 56	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 1𝑜) = suc 𝐴)
58		suceq 4139	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
59	57, 58	sylan9eq 2092	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → (𝐴 +𝑜 1𝑜) = suc 𝑦)
60		1onn 6093	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ ω
61		eleq2 2101	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1𝑜 → (∅ ∈ 𝑥 ↔ ∅ ∈ 1𝑜))
62		oveq2 5520	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1𝑜 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 1𝑜))
63	62	eqeq1d 2048	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1𝑜 → ((𝐴 +𝑜 𝑥) = suc 𝑦 ↔ (𝐴 +𝑜 1𝑜) = suc 𝑦))
64	61, 63	anbi12d 442	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1𝑜 → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦) ↔ (∅ ∈ 1𝑜 ∧ (𝐴 +𝑜 1𝑜) = suc 𝑦)))
65	64	rspcev 2656	. . . . . . . . . . . 12 ⊢ ((1𝑜 ∈ ω ∧ (∅ ∈ 1𝑜 ∧ (𝐴 +𝑜 1𝑜) = suc 𝑦)) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
66	60, 65	mpan 400	. . . . . . . . . . 11 ⊢ ((∅ ∈ 1𝑜 ∧ (𝐴 +𝑜 1𝑜) = suc 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
67	54, 59, 66	syl2anc 391	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦))
68	67	ex 108	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
69	68	ad2antlr 458	. . . . . . . 8 ⊢ (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
70	52, 69	jaod 637	. . . . . . 7 ⊢ (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → ((𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
71	25, 70	syl5 28	. . . . . 6 ⊢ (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦))) → (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))
72	71	exp31 346	. . . . 5 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑦)) → (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = suc 𝑦)))))
73	11, 16, 21, 24, 72	finds2 4324	. . . 4 ⊢ (𝑏 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝑏))))
74	6, 73	vtoclga 2619	. . 3 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵))))
75	74	impcom 116	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
76		peano1 4317	. . . . . . . . 9 ⊢ ∅ ∈ ω
77		nnaord 6082	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
78	76, 77	mp3an1 1219	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
79	78	ancoms 255	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥)))
80		nna0 6053	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
81	80	adantr 261	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 ∅) = 𝐴)
82	81	eleq1d 2106	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 ∅) ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴 ∈ (𝐴 +𝑜 𝑥)))
83	79, 82	bitrd 177	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ (𝐴 +𝑜 𝑥)))
84	83	anbi1d 438	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) ↔ (𝐴 ∈ (𝐴 +𝑜 𝑥) ∧ (𝐴 +𝑜 𝑥) = 𝐵)))
85		eleq2 2101	. . . . . 6 ⊢ ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +𝑜 𝑥) ↔ 𝐴 ∈ 𝐵))
86	85	biimpac 282	. . . . 5 ⊢ ((𝐴 ∈ (𝐴 +𝑜 𝑥) ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴 ∈ 𝐵)
87	84, 86	syl6bi 152	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
88	87	rexlimdva 2433	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
89	88	adantr 261	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
90	75, 89	impbid 120	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +𝑜 𝑥) = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  ∅c0 3224  Oncon0 4100  suc csuc 4102  ωcom 4313  (class class class)co 5512  1_𝑜c1o 5994   +_𝑜 coa 5998

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-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005

This theorem is referenced by:  nnawordex  6101  ltexpi  6435