Theorem nnaordex 6036

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 2098	. . . . . 6 ⊢ (𝑏 = B → (A ∈ 𝑏 ↔ A ∈ B))
2		eqeq2 2046	. . . . . . . 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 2321	. . . . . 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 2098	. . . . . 6 ⊢ (𝑏 = ∅ → (A ∈ 𝑏 ↔ A ∈ ∅))
8		eqeq2 2046	. . . . . . . 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 2321	. . . . . 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 2098	. . . . . 6 ⊢ (𝑏 = y → (A ∈ 𝑏 ↔ A ∈ y))
13		eqeq2 2046	. . . . . . . 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 2321	. . . . . 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 2098	. . . . . 6 ⊢ (𝑏 = suc y → (A ∈ 𝑏 ↔ A ∈ suc y))
18		eqeq2 2046	. . . . . . . 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 2321	. . . . . 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 3222	. . . . . . 7 ⊢ ¬ A ∈ ∅
23	22	pm2.21i 574	. . . . . 6 ⊢ (A ∈ ∅ → ∃x ∈ 𝜔 (∅ ∈ x ∧ (A +𝑜 x) = ∅))
24	23	a1i 9	. . . . 5 ⊢ (A ∈ 𝜔 → (A ∈ ∅ → ∃x ∈ 𝜔 (∅ ∈ x ∧ (A +𝑜 x) = ∅)))
25		elsuci 4106	. . . . . . 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 4261	. . . . . . . . . . . . . . 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 4112	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∈ x → ∅ ∈ suc x)
30	29	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ 𝜔 ∧ x ∈ 𝜔) → (∅ ∈ x → ∅ ∈ suc x))
31		nnasuc 5994	. . . . . . . . . . . . . . . . . 18 ⊢ ((A ∈ 𝜔 ∧ x ∈ 𝜔) → (A +𝑜 suc x) = suc (A +𝑜 x))
32		suceq 4105	. . . . . . . . . . . . . . . . . 18 ⊢ ((A +𝑜 x) = y → suc (A +𝑜 x) = suc y)
33	31, 32	sylan9eq 2089	. . . . . . . . . . . . . . . . 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 2098	. . . . . . . . . . . . . . . 16 ⊢ (z = suc x → (∅ ∈ z ↔ ∅ ∈ suc x))
38		oveq2 5463	. . . . . . . . . . . . . . . . 17 ⊢ (z = suc x → (A +𝑜 z) = (A +𝑜 suc x))
39	38	eqeq1d 2045	. . . . . . . . . . . . . . . 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 2650	. . . . . . . . . . . . . 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 2427	. . . . . . . . . . 11 ⊢ (A ∈ 𝜔 → (∃x ∈ 𝜔 (∅ ∈ x ∧ (A +𝑜 x) = y) → ∃z ∈ 𝜔 (∅ ∈ z ∧ (A +𝑜 z) = suc y)))
45		eleq2 2098	. . . . . . . . . . . . 13 ⊢ (z = x → (∅ ∈ z ↔ ∅ ∈ x))
46		oveq2 5463	. . . . . . . . . . . . . 14 ⊢ (z = x → (A +𝑜 z) = (A +𝑜 x))
47	46	eqeq1d 2045	. . . . . . . . . . . . 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 2528	. . . . . . . . . . 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 5962	. . . . . . . . . . . 12 ⊢ ∅ ∈ 1𝑜
54	53	a1i 9	. . . . . . . . . . 11 ⊢ ((A ∈ 𝜔 ∧ A = y) → ∅ ∈ 1𝑜)
55		nnon 4275	. . . . . . . . . . . . 13 ⊢ (A ∈ 𝜔 → A ∈ On)
56		oa1suc 5986	. . . . . . . . . . . . 13 ⊢ (A ∈ On → (A +𝑜 1𝑜) = suc A)
57	55, 56	syl 14	. . . . . . . . . . . 12 ⊢ (A ∈ 𝜔 → (A +𝑜 1𝑜) = suc A)
58		suceq 4105	. . . . . . . . . . . 12 ⊢ (A = y → suc A = suc y)
59	57, 58	sylan9eq 2089	. . . . . . . . . . 11 ⊢ ((A ∈ 𝜔 ∧ A = y) → (A +𝑜 1𝑜) = suc y)
60		1onn 6029	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ 𝜔
61		eleq2 2098	. . . . . . . . . . . . . 14 ⊢ (x = 1𝑜 → (∅ ∈ x ↔ ∅ ∈ 1𝑜))
62		oveq2 5463	. . . . . . . . . . . . . . 15 ⊢ (x = 1𝑜 → (A +𝑜 x) = (A +𝑜 1𝑜))
63	62	eqeq1d 2045	. . . . . . . . . . . . . 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 2650	. . . . . . . . . . . 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 636	. . . . . . 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 4267	. . . 4 ⊢ (𝑏 ∈ 𝜔 → (A ∈ 𝜔 → (A ∈ 𝑏 → ∃x ∈ 𝜔 (∅ ∈ x ∧ (A +𝑜 x) = 𝑏))))
74	6, 73	vtoclga 2613	. . 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 4260	. . . . . . . . 9 ⊢ ∅ ∈ 𝜔
77		nnaord 6018	. . . . . . . . 9 ⊢ ((∅ ∈ 𝜔 ∧ x ∈ 𝜔 ∧ A ∈ 𝜔) → (∅ ∈ x ↔ (A +𝑜 ∅) ∈ (A +𝑜 x)))
78	76, 77	mp3an1 1218	. . . . . . . 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 5992	. . . . . . . . 9 ⊢ (A ∈ 𝜔 → (A +𝑜 ∅) = A)
81	80	adantr 261	. . . . . . . 8 ⊢ ((A ∈ 𝜔 ∧ x ∈ 𝜔) → (A +𝑜 ∅) = A)
82	81	eleq1d 2103	. . . . . . 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 2098	. . . . . 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 2427	. . 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 628   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  ∅c0 3218  Oncon0 4066  suc csuc 4068  𝜔com 4256  (class class class)co 5455  1_𝑜c1o 5933   +_𝑜 coa 5937

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944

This theorem is referenced by:  nnawordex  6037  ltexpi  6321