Theorem nnaordex 6495

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 2230	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝐵))
2		eqeq2 2175	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝐵))
3	2	anbi2d 460	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
4	3	rexbidv 2467	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
5	1, 4	imbi12d 233	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
6	5	imbi2d 229	. . . 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 2230	. . . . . 6 ⊢ (𝑏 = ∅ → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ ∅))
8		eqeq2 2175	. . . . . . . 8 ⊢ (𝑏 = ∅ → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = ∅))
9	8	anbi2d 460	. . . . . . 7 ⊢ (𝑏 = ∅ → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
10	9	rexbidv 2467	. . . . . 6 ⊢ (𝑏 = ∅ → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
11	7, 10	imbi12d 233	. . . . 5 ⊢ (𝑏 = ∅ → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))))
12		eleq2 2230	. . . . . 6 ⊢ (𝑏 = 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ 𝑦))
13		eqeq2 2175	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝑦))
14	13	anbi2d 460	. . . . . . 7 ⊢ (𝑏 = 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
15	14	rexbidv 2467	. . . . . 6 ⊢ (𝑏 = 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
16	12, 15	imbi12d 233	. . . . 5 ⊢ (𝑏 = 𝑦 → ((𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))))
17		eleq2 2230	. . . . . 6 ⊢ (𝑏 = suc 𝑦 → (𝐴 ∈ 𝑏 ↔ 𝐴 ∈ suc 𝑦))
18		eqeq2 2175	. . . . . . . 8 ⊢ (𝑏 = suc 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = suc 𝑦))
19	18	anbi2d 460	. . . . . . 7 ⊢ (𝑏 = suc 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
20	19	rexbidv 2467	. . . . . 6 ⊢ (𝑏 = suc 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
21	17, 20	imbi12d 233	. . . . 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 3413	. . . . . . 7 ⊢ ¬ 𝐴 ∈ ∅
23	22	pm2.21i 636	. . . . . 6 ⊢ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))
24	23	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
25		elsuci 4381	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝑦 → (𝐴 ∈ 𝑦 ∨ 𝐴 = 𝑦))
26		simpr 109	. . . . . . . . 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 4572	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
28	27	ad2antlr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → suc 𝑥 ∈ ω)
29		elelsuc 4387	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥)
30	29	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥))
31		nnasuc 6444	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
32		suceq 4380	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 +o 𝑥) = 𝑦 → suc (𝐴 +o 𝑥) = suc 𝑦)
33	31, 32	sylan9eq 2219	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (𝐴 +o 𝑥) = 𝑦) → (𝐴 +o suc 𝑥) = suc 𝑦)
34	33	ex 114	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝑦 → (𝐴 +o suc 𝑥) = suc 𝑦))
35	30, 34	anim12d 333	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
36	35	imp 123	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦))
37		eleq2 2230	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ suc 𝑥))
38		oveq2 5850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = suc 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑥))
39	38	eqeq1d 2174	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o suc 𝑥) = suc 𝑦))
40	37, 39	anbi12d 465	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
41	40	rspcev 2830	. . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦))
43	42	ex 114	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
44	43	rexlimdva 2583	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
45		eleq2 2230	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑥))
46		oveq2 5850	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o 𝑥))
47	46	eqeq1d 2174	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o 𝑥) = suc 𝑦))
48	45, 47	anbi12d 465	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
49	48	cbvrexv 2693	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
50	44, 49	syl6ib 160	. . . . . . . . . 10 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
51	50	ad2antlr 481	. . . . . . . . 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 6408	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1o
54	53	a1i 9	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∅ ∈ 1o)
55		nnon 4587	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
56		oa1suc 6435	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)
57	55, 56	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ω → (𝐴 +o 1o) = suc 𝐴)
58		suceq 4380	. . . . . . . . . . . 12 ⊢ (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
59	57, 58	sylan9eq 2219	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → (𝐴 +o 1o) = suc 𝑦)
60		1onn 6488	. . . . . . . . . . . 12 ⊢ 1o ∈ ω
61		eleq2 2230	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1o → (∅ ∈ 𝑥 ↔ ∅ ∈ 1o))
62		oveq2 5850	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1o → (𝐴 +o 𝑥) = (𝐴 +o 1o))
63	62	eqeq1d 2174	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1o → ((𝐴 +o 𝑥) = suc 𝑦 ↔ (𝐴 +o 1o) = suc 𝑦))
64	61, 63	anbi12d 465	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1o → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦) ↔ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)))
65	64	rspcev 2830	. . . . . . . . . . . 12 ⊢ ((1o ∈ ω ∧ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
66	60, 65	mpan 421	. . . . . . . . . . 11 ⊢ ((∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
67	54, 59, 66	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
68	67	ex 114	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
69	68	ad2antlr 481	. . . . . . . 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 707	. . . . . . 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 362	. . . . 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 4578	. . . 4 ⊢ (𝑏 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏))))
74	6, 73	vtoclga 2792	. . 3 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
75	74	impcom 124	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
76		peano1 4571	. . . . . . . . 9 ⊢ ∅ ∈ ω
77		nnaord 6477	. . . . . . . . 9 ⊢ ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
78	76, 77	mp3an1 1314	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
79	78	ancoms 266	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
80		nna0 6442	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
81	80	adantr 274	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o ∅) = 𝐴)
82	81	eleq1d 2235	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o ∅) ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
83	79, 82	bitrd 187	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
84	83	anbi1d 461	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) ↔ (𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵)))
85		eleq2 2230	. . . . . 6 ⊢ ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ 𝐵))
86	85	biimpac 296	. . . . 5 ⊢ ((𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵)
87	84, 86	syl6bi 162	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
88	87	rexlimdva 2583	. . 3 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
89	88	adantr 274	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴 ∈ 𝐵))
90	75, 89	impbid 128	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  ∅c0 3409  Oncon0 4341  suc csuc 4343  ωcom 4567  (class class class)co 5842  1_oc1o 6377   +_o coa 6381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388

This theorem is referenced by:  nnawordex  6496  ltexpi  7278