Theorem pssnnOLD 8872

Description: Obsolete version of pssnn 8824 as of 31-Jul-2024. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pssnnOLD	⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pssnnOLD

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

Step	Hyp	Ref	Expression
1		pssss 3996	. . . 4 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
2		ssexg 5201	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ V)
3	1, 2	sylan 583	. . 3 ⊢ ((𝐵 ⊊ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ V)
4	3	ancoms 462	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ∈ V)
5		psseq2 3989	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ ∅))
6		rexeq 3310	. . . . . . . 8 ⊢ (𝑧 = ∅ → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥))
7	5, 6	imbi12d 348	. . . . . . 7 ⊢ (𝑧 = ∅ → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)))
8	7	albidv 1928	. . . . . 6 ⊢ (𝑧 = ∅ → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)))
9		psseq2 3989	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝑦))
10		rexeq 3310	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥))
11	9, 10	imbi12d 348	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)))
12	11	albidv 1928	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)))
13		psseq2 3989	. . . . . . . 8 ⊢ (𝑧 = suc 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ suc 𝑦))
14		rexeq 3310	. . . . . . . 8 ⊢ (𝑧 = suc 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
15	13, 14	imbi12d 348	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
16	15	albidv 1928	. . . . . 6 ⊢ (𝑧 = suc 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
17		psseq2 3989	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝐴))
18		rexeq 3310	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))
19	17, 18	imbi12d 348	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)))
20	19	albidv 1928	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)))
21		npss0 4346	. . . . . . . 8 ⊢ ¬ 𝑤 ⊊ ∅
22	21	pm2.21i 119	. . . . . . 7 ⊢ (𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)
23	22	ax-gen 1803	. . . . . 6 ⊢ ∀𝑤(𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)
24		nfv 1922	. . . . . . 7 ⊢ Ⅎ𝑤 𝑦 ∈ ω
25		nfa1 2154	. . . . . . 7 ⊢ Ⅎ𝑤∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)
26		elequ1 2119	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
27	26	biimpcd 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑤 → (𝑧 = 𝑦 → 𝑦 ∈ 𝑤))
28	27	con3d 155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑤 → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦))
29	28	adantl 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦))
30		pssss 3996	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ⊊ suc 𝑦 → 𝑤 ⊆ suc 𝑦)
31	30	sseld 3886	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ⊊ suc 𝑦 → (𝑧 ∈ 𝑤 → 𝑧 ∈ suc 𝑦))
32		elsuci 6257	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ suc 𝑦 → (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦))
33	32	ord 864	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ suc 𝑦 → (¬ 𝑧 ∈ 𝑦 → 𝑧 = 𝑦))
34	33	con1d 147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ suc 𝑦 → (¬ 𝑧 = 𝑦 → 𝑧 ∈ 𝑦))
35	31, 34	syl6 35	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ⊊ suc 𝑦 → (𝑧 ∈ 𝑤 → (¬ 𝑧 = 𝑦 → 𝑧 ∈ 𝑦)))
36	35	imp 410	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑧 = 𝑦 → 𝑧 ∈ 𝑦))
37	29, 36	syld 47	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑦 ∈ 𝑤 → 𝑧 ∈ 𝑦))
38	37	impancom 455	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) → (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦))
39	38	ssrdv 3893	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) → 𝑤 ⊆ 𝑦)
40	39	anim1i 618	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦))
41		dfpss2 3986	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ⊊ 𝑦 ↔ (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦))
42	40, 41	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → 𝑤 ⊊ 𝑦)
43		elelsuc 6263	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ suc 𝑦)
44	43	anim1i 618	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑤 ≈ 𝑥) → (𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑥))
45	44	reximi2 3157	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)
46	42, 45	imim12i 62	. . . . . . . . . . . . 13 ⊢ ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
47	46	exp4c 436	. . . . . . . . . . . 12 ⊢ ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
48	47	sps 2184	. . . . . . . . . . 11 ⊢ (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
49	48	adantl 485	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
50	49	com4t 93	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
51		anidm 568	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑤 ⊊ suc 𝑦) ↔ 𝑤 ⊊ suc 𝑦)
52		ssdif 4040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ⊆ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦}))
53		nnord 7630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → Ord 𝑦)
54		orddif 6284	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑦 → 𝑦 = (suc 𝑦 ∖ {𝑦}))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → 𝑦 = (suc 𝑦 ∖ {𝑦}))
56	55	sseq2d 3919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦})))
57	52, 56	syl5ibr 249	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ω → (𝑤 ⊆ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ 𝑦))
58	30, 57	syl5 34	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ω → (𝑤 ⊊ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ 𝑦))
59		pssnel 4371	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ⊊ suc 𝑦 → ∃𝑧(𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤))
60		eleq2 2819	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∖ {𝑦}) = 𝑦 → (𝑧 ∈ (𝑤 ∖ {𝑦}) ↔ 𝑧 ∈ 𝑦))
61		eldifi 4027	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝑤 ∖ {𝑦}) → 𝑧 ∈ 𝑤)
62	60, 61	syl6bir 257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∖ {𝑦}) = 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
63	62	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
64		eleq1a 2826	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝑤 → (𝑧 = 𝑦 → 𝑧 ∈ 𝑤))
65	33, 64	sylan9r 512	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → (¬ 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
66	65	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → (¬ 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
67	63, 66	pm2.61d 182	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → 𝑧 ∈ 𝑤)
68	67	ex 416	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → ((𝑤 ∖ {𝑦}) = 𝑦 → 𝑧 ∈ 𝑤))
69	68	con3d 155	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → (¬ 𝑧 ∈ 𝑤 → ¬ (𝑤 ∖ {𝑦}) = 𝑦))
70	69	expimpd 457	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑤 → ((𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤) → ¬ (𝑤 ∖ {𝑦}) = 𝑦))
71	70	exlimdv 1941	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑤 → (∃𝑧(𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤) → ¬ (𝑤 ∖ {𝑦}) = 𝑦))
72	59, 71	syl5 34	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑤 → (𝑤 ⊊ suc 𝑦 → ¬ (𝑤 ∖ {𝑦}) = 𝑦))
73	58, 72	im2anan9r 624	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑤 ⊊ suc 𝑦 ∧ 𝑤 ⊊ suc 𝑦) → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦)))
74	51, 73	syl5bir 246	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦)))
75		dfpss2 3986	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 ↔ ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦))
76	74, 75	syl6ibr 255	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊊ 𝑦))
77		psseq1 3988	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑤 ⊊ 𝑦 ↔ 𝑧 ⊊ 𝑦))
78		breq1 5042	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑧 ≈ 𝑥))
79	78	rexbidv 3206	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥))
80	77, 79	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ (𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥)))
81	80	cbvalvw 2046	. . . . . . . . . . . . 13 ⊢ (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ ∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥))
82		vex 3402	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
83	82	difexi 5206	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {𝑦}) ∈ V
84		psseq1 3988	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ⊊ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊊ 𝑦))
85		breq1 5042	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ≈ 𝑥 ↔ (𝑤 ∖ {𝑦}) ≈ 𝑥))
86	85	rexbidv 3206	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
87	84, 86	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → ((𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) ↔ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥)))
88	83, 87	spcv 3510	. . . . . . . . . . . . 13 ⊢ (∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) → ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
89	81, 88	sylbi 220	. . . . . . . . . . . 12 ⊢ (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
90	76, 89	sylan9 511	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
91		ordsucelsuc 7579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Ord 𝑦 → (𝑥 ∈ 𝑦 ↔ suc 𝑥 ∈ suc 𝑦))
92	91	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑦 → (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦))
93	53, 92	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦))
94	93	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦))
95	94	adantrd 495	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → suc 𝑥 ∈ suc 𝑦))
96		elnn 7633	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω)
97		snex 5309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑦, 𝑥⟩} ∈ V
98		vex 3402	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
99		vex 3402	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑥 ∈ V
100	98, 99	f1osn 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑦, 𝑥⟩}:{𝑦}–1-1-onto→{𝑥}
101		f1oen3g 8622	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({⟨𝑦, 𝑥⟩} ∈ V ∧ {⟨𝑦, 𝑥⟩}:{𝑦}–1-1-onto→{𝑥}) → {𝑦} ≈ {𝑥})
102	97, 100, 101	mp2an 692	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑦} ≈ {𝑥}
103	102	jctr 528	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∖ {𝑦}) ≈ 𝑥 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥}))
104		nnord 7630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ω → Ord 𝑥)
105		orddisj 6229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
106	104, 105	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
107		incom 4101	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝑦} ∩ (𝑤 ∖ {𝑦})) = ((𝑤 ∖ {𝑦}) ∩ {𝑦})
108		disjdif 4372	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝑦} ∩ (𝑤 ∖ {𝑦})) = ∅
109	107, 108	eqtr3i 2761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅
110	106, 109	jctil 523	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ω → (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅))
111		unen 8701	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥}) ∧ (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥}))
112	103, 110, 111	syl2an 599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥}))
113		difsnid 4709	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝑤 → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) = 𝑤)
114	113	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑤 → 𝑤 = ((𝑤 ∖ {𝑦}) ∪ {𝑦}))
115		df-suc 6197	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
116	115	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑤 → suc 𝑥 = (𝑥 ∪ {𝑥}))
117	114, 116	breq12d 5052	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝑤 → (𝑤 ≈ suc 𝑥 ↔ ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥})))
118	112, 117	syl5ibr 249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑤 ≈ suc 𝑥))
119	96, 118	sylan2i 609	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω)) → 𝑤 ≈ suc 𝑥))
120	119	exp4d 437	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝑤 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 → (𝑥 ∈ 𝑦 → (𝑦 ∈ ω → 𝑤 ≈ suc 𝑥))))
121	120	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑤 → (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 → 𝑤 ≈ suc 𝑥))))
122	121	imp4b 425	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → 𝑤 ≈ suc 𝑥))
123	95, 122	jcad 516	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → (suc 𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ suc 𝑥)))
124		breq2 5043	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (𝑤 ≈ 𝑧 ↔ 𝑤 ≈ suc 𝑥))
125	124	rspcev 3527	. . . . . . . . . . . . . . 15 ⊢ ((suc 𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ suc 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧)
126	123, 125	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧))
127	126	exlimdv 1941	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧))
128		df-rex 3057	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥))
129		breq2 5043	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑧))
130	129	cbvrexvw 3349	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥 ↔ ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧)
131	127, 128, 130	3imtr4g 299	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
132	131	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
133	90, 132	syld 47	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
134	133	expl 461	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑤 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
135	82	eqelsuc 6272	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → 𝑤 ∈ suc 𝑦)
136	82	enref 8639	. . . . . . . . . . 11 ⊢ 𝑤 ≈ 𝑤
137		breq2 5043	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑤))
138	137	rspcev 3527	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑤) → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)
139	135, 136, 138	sylancl 589	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)
140	139	2a1d 26	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
141	50, 134, 140	pm2.61ii 186	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
142	141	ex 416	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
143	24, 25, 142	alrimd 2215	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → ∀𝑤(𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
144	8, 12, 16, 20, 23, 143	finds 7654	. . . . 5 ⊢ (𝐴 ∈ ω → ∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))
145		psseq1 3988	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑤 ⊊ 𝐴 ↔ 𝐵 ⊊ 𝐴))
146		breq1 5042	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → (𝑤 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥))
147	146	rexbidv 3206	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))
148	145, 147	imbi12d 348	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥) ↔ (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
149	148	spcgv 3501	. . . . 5 ⊢ (𝐵 ∈ V → (∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥) → (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
150	144, 149	syl5 34	. . . 4 ⊢ (𝐵 ∈ V → (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
151	150	com3l 89	. . 3 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → (𝐵 ∈ V → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
152	151	imp 410	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (𝐵 ∈ V → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))
153	4, 152	mpd 15	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2112  ∃wrex 3052  Vcvv 3398   ∖ cdif 3850   ∪ cun 3851   ∩ cin 3852   ⊆ wss 3853   ⊊ wpss 3854  ∅c0 4223  {csn 4527  ⟨cop 4533   class class class wbr 5039  Ord word 6190  suc csuc 6193  –1-1-onto→wf1o 6357  ωcom 7622   ≈ cen 8601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-om 7623  df-en 8605

This theorem is referenced by: (None)