Theorem nnaordex 6581

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 2257	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2203	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝐵))
3	2	anbi2d 464	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
4	3	rexbidv 2495	. . . . . 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 2257	. . . . . 6 ⊢ (𝑏 = ∅ → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ ∅))
8		eqeq2 2203	. . . . . . . 8 ⊢ (𝑏 = ∅ → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = ∅))
9	8	anbi2d 464	. . . . . . 7 ⊢ (𝑏 = ∅ → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
10	9	rexbidv 2495	. . . . . 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 2257	. . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝑦))
13		eqeq2 2203	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝑦))
14	13	anbi2d 464	. . . . . . 7 ⊢ (𝑏 = 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
15	14	rexbidv 2495	. . . . . 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 2257	. . . . . 6 ⊢ (𝑏 = suc 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ suc 𝑦))
18		eqeq2 2203	. . . . . . . 8 ⊢ (𝑏 = suc 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = suc 𝑦))
19	18	anbi2d 464	. . . . . . 7 ⊢ (𝑏 = suc 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
20	19	rexbidv 2495	. . . . . 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 3450	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
23	22	pm2.21i 647	. . . . . 6 ⊢ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))
24	23	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
25		elsuci 4434	. . . . . . 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 4627	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
28	27	ad2antlr 489	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → suc 𝑥 ∈ ω)
29		elelsuc 4440	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥)
30	29	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥))
31		nnasuc 6529	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
32		suceq 4433	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 +o 𝑥) = 𝑦 → suc (𝐴 +o 𝑥) = suc 𝑦)
33	31, 32	sylan9eq 2246	. . . . . . . . . . . . . . . . 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 2257	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ suc 𝑥))
38		oveq2 5926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑥))
39	38	eqeq1d 2202	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o suc 𝑥) = suc 𝑦))
40	37, 39	anbi12d 473	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
41	40	rspcev 2864	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 2611	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
45		eleq2 2257	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑥))
46		oveq2 5926	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o 𝑥))
47	46	eqeq1d 2202	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o 𝑥) = suc 𝑦))
48	45, 47	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
49	48	cbvrexv 2727	. . . . . . . . . . 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 489	. . . . . . . . 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 6493	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1o
54	53	a1i 9	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∅ ∈ 1o)
55		nnon 4642	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
56		oa1suc 6520	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)
57	55, 56	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (𝐴 +o 1o) = suc 𝐴)
58		suceq 4433	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
59	57, 58	sylan9eq 2246	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → (𝐴 +o 1o) = suc 𝑦)
60		1onn 6573	. . . . . . . . . . . 12 ⊢ 1o ∈ ω
61		eleq2 2257	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1o → (∅ ∈ 𝑥 ↔ ∅ ∈ 1o))
62		oveq2 5926	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1o → (𝐴 +o 𝑥) = (𝐴 +o 1o))
63	62	eqeq1d 2202	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1o → ((𝐴 +o 𝑥) = suc 𝑦 ↔ (𝐴 +o 1o) = suc 𝑦))
64	61, 63	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1o → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦) ↔ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)))
65	64	rspcev 2864	. . . . . . . . . . . 12 ⊢ ((1o ∈ ω ∧ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
66	60, 65	mpan 424	. . . . . . . . . . 11 ⊢ ((∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
67	54, 59, 66	syl2anc 411	. . . . . . . . . 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 489	. . . . . . . 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 718	. . . . . . 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 4633	. . . 4 ⊢ (𝑏 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏))))
74	6, 73	vtoclga 2826	. . 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 4626	. . . . . . . . 9 ⊢ ∅ ∈ ω
77		nnaord 6562	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
78	76, 77	mp3an1 1335	. . . . . . . 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 6527	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
81	80	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o ∅) = 𝐴)
82	81	eleq1d 2262	. . . . . . 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 465	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) ↔ (𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵)))
85		eleq2 2257	. . . . . 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 2611	. . 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 709   = wceq 1364   ∈ wcel 2164  ∃wrex 2473  ∅c0 3446  Oncon0 4394  suc csuc 4396  ωcom 4622  (class class class)co 5918  1_oc1o 6462   +_o coa 6466

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473

This theorem is referenced by:  nnawordex  6582  ltexpi  7397