Theorem nnaordex 6791

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 2302	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2248	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝐵))
3	2	anbi2d 468	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
4	3	rexbidv 2551	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
5	1, 4	imbi12d 234	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
6	5	imbi2d 230	. . . 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 2302	. . . . . 6 ⊢ (𝑏 = ∅ → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ ∅))
8		eqeq2 2248	. . . . . . . 8 ⊢ (𝑏 = ∅ → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = ∅))
9	8	anbi2d 468	. . . . . . 7 ⊢ (𝑏 = ∅ → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
10	9	rexbidv 2551	. . . . . 6 ⊢ (𝑏 = ∅ → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
11	7, 10	imbi12d 234	. . . . 5 ⊢ (𝑏 = ∅ → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))))
12		eleq2 2302	. . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝑦))
13		eqeq2 2248	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝑦))
14	13	anbi2d 468	. . . . . . 7 ⊢ (𝑏 = 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
15	14	rexbidv 2551	. . . . . 6 ⊢ (𝑏 = 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
16	12, 15	imbi12d 234	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))))
17		eleq2 2302	. . . . . 6 ⊢ (𝑏 = suc 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ suc 𝑦))
18		eqeq2 2248	. . . . . . . 8 ⊢ (𝑏 = suc 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = suc 𝑦))
19	18	anbi2d 468	. . . . . . 7 ⊢ (𝑏 = suc 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
20	19	rexbidv 2551	. . . . . 6 ⊢ (𝑏 = suc 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
21	17, 20	imbi12d 234	. . . . 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 3525	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
23	22	pm2.21i 655	. . . . . 6 ⊢ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))
24	23	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
25		elsuci 4543	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
26		simpr 110	. . . . . . . . 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 4737	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
28	27	ad2antlr 493	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → suc 𝑥 ∈ ω)
29		elelsuc 4549	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥)
30	29	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥))
31		nnasuc 6739	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
32		suceq 4542	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 +o 𝑥) = 𝑦 → suc (𝐴 +o 𝑥) = suc 𝑦)
33	31, 32	sylan9eq 2291	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (𝐴 +o 𝑥) = 𝑦) → (𝐴 +o suc 𝑥) = suc 𝑦)
34	33	ex 115	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝑦 → (𝐴 +o suc 𝑥) = suc 𝑦))
35	30, 34	anim12d 335	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
36	35	imp 124	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦))
37		eleq2 2302	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ suc 𝑥))
38		oveq2 6083	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑥))
39	38	eqeq1d 2247	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o suc 𝑥) = suc 𝑦))
40	37, 39	anbi12d 477	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
41	40	rspcev 2929	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦))
43	42	ex 115	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
44	43	rexlimdva 2668	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
45		eleq2 2302	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑥))
46		oveq2 6083	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o 𝑥))
47	46	eqeq1d 2247	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o 𝑥) = suc 𝑦))
48	45, 47	anbi12d 477	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
49	48	cbvrexv 2787	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
50	44, 49	imbitrdi 161	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
51	50	ad2antlr 493	. . . . . . . . 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 45	. . . . . . . 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 6703	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1o
54	53	a1i 9	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∅ ∈ 1o)
55		nnon 4752	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
56		oa1suc 6730	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)
57	55, 56	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (𝐴 +o 1o) = suc 𝐴)
58		suceq 4542	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
59	57, 58	sylan9eq 2291	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → (𝐴 +o 1o) = suc 𝑦)
60		1onn 6783	. . . . . . . . . . . 12 ⊢ 1o ∈ ω
61		eleq2 2302	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1o → (∅ ∈ 𝑥 ↔ ∅ ∈ 1o))
62		oveq2 6083	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1o → (𝐴 +o 𝑥) = (𝐴 +o 1o))
63	62	eqeq1d 2247	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1o → ((𝐴 +o 𝑥) = suc 𝑦 ↔ (𝐴 +o 1o) = suc 𝑦))
64	61, 63	anbi12d 477	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1o → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦) ↔ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)))
65	64	rspcev 2929	. . . . . . . . . . . 12 ⊢ ((1o ∈ ω ∧ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
66	60, 65	mpan 428	. . . . . . . . . . 11 ⊢ ((∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
67	54, 59, 66	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
68	67	ex 115	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
69	68	ad2antlr 493	. . . . . . . 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 729	. . . . . . 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 32	. . . . . 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 364	. . . . 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 4743	. . . 4 ⊢ (𝑏 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏))))
74	6, 73	vtoclga 2889	. . 3 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
75	74	impcom 125	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
76		peano1 4736	. . . . . . . . 9 ⊢ ∅ ∈ ω
77		nnaord 6772	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
78	76, 77	mp3an1 1365	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
79	78	ancoms 268	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
80		nna0 6737	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
81	80	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o ∅) = 𝐴)
82	81	eleq1d 2307	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o ∅) ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
83	79, 82	bitrd 188	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
84	83	anbi1d 469	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) ↔ (𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵)))
85		eleq2 2302	. . . . . 6 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ 𝐵))
86	85	biimpac 298	. . . . 5 ⊢ ((𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵)
87	84, 86	biimtrdi 163	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
88	87	rexlimdva 2668	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
89	88	adantr 276	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
90	75, 89	impbid 129	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ∅c0 3520  Oncon0 4503  suc csuc 4505  ωcom 4732  (class class class)co 6075  1_oc1o 6670   +_o coa 6674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681

This theorem is referenced by:  nnawordex  6792  ltexpi  7694