Theorem pssnn 8951

Description: A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5288. (Revised by BTernaryTau, 31-Jul-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝐵   𝑥,𝐴

Proof of Theorem pssnn

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

Step	Hyp	Ref	Expression
1		pssss 4030	. . . 4 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
2		ssexg 5247	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ V)
3	1, 2	sylan 580	. . 3 ⊢ ((𝐵 ⊊ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ V)
4	3	ancoms 459	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ∈ V)
5		psseq2 4023	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ ∅))
6		rexeq 3343	. . . . . . . 8 ⊢ (𝑧 = ∅ → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥))
7	5, 6	imbi12d 345	. . . . . . 7 ⊢ (𝑧 = ∅ → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)))
8	7	albidv 1923	. . . . . 6 ⊢ (𝑧 = ∅ → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)))
9		psseq2 4023	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝑦))
10		rexeq 3343	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥))
11	9, 10	imbi12d 345	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)))
12	11	albidv 1923	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)))
13		psseq2 4023	. . . . . . . 8 ⊢ (𝑧 = suc 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ suc 𝑦))
14		rexeq 3343	. . . . . . . 8 ⊢ (𝑧 = suc 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
15	13, 14	imbi12d 345	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
16	15	albidv 1923	. . . . . 6 ⊢ (𝑧 = suc 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
17		psseq2 4023	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝐴))
18		rexeq 3343	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))
19	17, 18	imbi12d 345	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)))
20	19	albidv 1923	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)))
21		npss0 4379	. . . . . . . 8 ⊢ ¬ 𝑤 ⊊ ∅
22	21	pm2.21i 119	. . . . . . 7 ⊢ (𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)
23	22	ax-gen 1798	. . . . . 6 ⊢ ∀𝑤(𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)
24		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑤 𝑦 ∈ ω
25		nfa1 2148	. . . . . . 7 ⊢ Ⅎ𝑤∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)
26		elequ1 2113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
27	26	biimpcd 248	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑤 → (𝑧 = 𝑦 → 𝑦 ∈ 𝑤))
28	27	con3d 152	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑤 → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦))
29	28	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦))
30		pssss 4030	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ⊊ suc 𝑦 → 𝑤 ⊆ suc 𝑦)
31	30	sseld 3920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ⊊ suc 𝑦 → (𝑧 ∈ 𝑤 → 𝑧 ∈ suc 𝑦))
32		elsuci 6332	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ suc 𝑦 → (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦))
33	32	ord 861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ suc 𝑦 → (¬ 𝑧 ∈ 𝑦 → 𝑧 = 𝑦))
34	33	con1d 145	. . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) → (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦))
39	38	ssrdv 3927	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) → 𝑤 ⊆ 𝑦)
40	39	anim1i 615	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦))
41		dfpss2 4020	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ⊊ 𝑦 ↔ (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦))
42	40, 41	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → 𝑤 ⊊ 𝑦)
43		elelsuc 6338	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ suc 𝑦)
44	43	anim1i 615	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑤 ≈ 𝑥) → (𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑥))
45	44	reximi2 3175	. . . . . . . . . . . . . 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 433	. . . . . . . . . . . 12 ⊢ ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
48	47	sps 2178	. . . . . . . . . . 11 ⊢ (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → (¬ 𝑦 ∈ 𝑤 → (¬ 𝑤 = 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
49	48	adantl 482	. . . . . . . . . 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 565	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑤 ⊊ suc 𝑦) ↔ 𝑤 ⊊ suc 𝑦)
52		ssdif 4074	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ⊆ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦}))
53		nnord 7720	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → Ord 𝑦)
54		orddif 6359	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑦 → 𝑦 = (suc 𝑦 ∖ {𝑦}))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → 𝑦 = (suc 𝑦 ∖ {𝑦}))
56	55	sseq2d 3953	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦})))
57	52, 56	syl5ibr 245	. . . . . . . . . . . . . . . 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 4404	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ⊊ suc 𝑦 → ∃𝑧(𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤))
60		eleq2 2827	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∖ {𝑦}) = 𝑦 → (𝑧 ∈ (𝑤 ∖ {𝑦}) ↔ 𝑧 ∈ 𝑦))
61		eldifi 4061	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝑤 ∖ {𝑦}) → 𝑧 ∈ 𝑤)
62	60, 61	syl6bir 253	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∖ {𝑦}) = 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
63	62	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
64		eleq1a 2834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝑤 → (𝑧 = 𝑦 → 𝑧 ∈ 𝑤))
65	33, 64	sylan9r 509	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → (¬ 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
66	65	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → (¬ 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
67	63, 66	pm2.61d 179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) ∧ (𝑤 ∖ {𝑦}) = 𝑦) → 𝑧 ∈ 𝑤)
68	67	ex 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → ((𝑤 ∖ {𝑦}) = 𝑦 → 𝑧 ∈ 𝑤))
69	68	con3d 152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑧 ∈ suc 𝑦) → (¬ 𝑧 ∈ 𝑤 → ¬ (𝑤 ∖ {𝑦}) = 𝑦))
70	69	expimpd 454	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑤 → ((𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤) → ¬ (𝑤 ∖ {𝑦}) = 𝑦))
71	70	exlimdv 1936	. . . . . . . . . . . . . . . 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 621	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑤 ⊊ suc 𝑦 ∧ 𝑤 ⊊ suc 𝑦) → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦)))
74	51, 73	syl5bir 242	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦)))
75		dfpss2 4020	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 ↔ ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦))
76	74, 75	syl6ibr 251	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊊ 𝑦))
77		psseq1 4022	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑤 ⊊ 𝑦 ↔ 𝑧 ⊊ 𝑦))
78		breq1 5077	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑧 ≈ 𝑥))
79	78	rexbidv 3226	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥))
80	77, 79	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ (𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥)))
81	80	cbvalvw 2039	. . . . . . . . . . . . 13 ⊢ (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ ∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥))
82		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
83	82	difexi 5252	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {𝑦}) ∈ V
84		psseq1 4022	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ⊊ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊊ 𝑦))
85		breq1 5077	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ≈ 𝑥 ↔ (𝑤 ∖ {𝑦}) ≈ 𝑥))
86	85	rexbidv 3226	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
87	84, 86	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → ((𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) ↔ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥)))
88	83, 87	spcv 3544	. . . . . . . . . . . . 13 ⊢ (∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) → ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
89	81, 88	sylbi 216	. . . . . . . . . . . 12 ⊢ (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
90	76, 89	sylan9 508	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
91		ordsucelsuc 7669	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Ord 𝑦 → (𝑥 ∈ 𝑦 ↔ suc 𝑥 ∈ suc 𝑦))
92	91	biimpd 228	. . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦))
95	94	adantrd 492	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → suc 𝑥 ∈ suc 𝑦))
96		elnn 7723	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω)
97		snex 5354	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑦, 𝑥⟩} ∈ V
98		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
99		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑥 ∈ V
100	98, 99	f1osn 6756	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑦, 𝑥⟩}:{𝑦}–1-1-onto→{𝑥}
101		f1oen3g 8754	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({⟨𝑦, 𝑥⟩} ∈ V ∧ {⟨𝑦, 𝑥⟩}:{𝑦}–1-1-onto→{𝑥}) → {𝑦} ≈ {𝑥})
102	97, 100, 101	mp2an 689	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑦} ≈ {𝑥}
103	102	jctr 525	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∖ {𝑦}) ≈ 𝑥 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥}))
104		nnord 7720	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ω → Ord 𝑥)
105		orddisj 6304	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
106	104, 105	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
107		disjdifr 4406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅
108	106, 107	jctil 520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ω → (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅))
109		unen 8836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥}) ∧ (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥}))
110	103, 108, 109	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥}))
111		difsnid 4743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝑤 → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) = 𝑤)
112	111	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑤 → 𝑤 = ((𝑤 ∖ {𝑦}) ∪ {𝑦}))
113		df-suc 6272	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
114	113	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑤 → suc 𝑥 = (𝑥 ∪ {𝑥}))
115	112, 114	breq12d 5087	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝑤 → (𝑤 ≈ suc 𝑥 ↔ ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥})))
116	110, 115	syl5ibr 245	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑤 ≈ suc 𝑥))
117	96, 116	sylan2i 606	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω)) → 𝑤 ≈ suc 𝑥))
118	117	exp4d 434	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝑤 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 → (𝑥 ∈ 𝑦 → (𝑦 ∈ ω → 𝑤 ≈ suc 𝑥))))
119	118	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑤 → (𝑦 ∈ ω → (𝑥 ∈ 𝑦 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 → 𝑤 ≈ suc 𝑥))))
120	119	imp4b 422	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → 𝑤 ≈ suc 𝑥))
121	95, 120	jcad 513	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → (suc 𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ suc 𝑥)))
122		breq2 5078	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (𝑤 ≈ 𝑧 ↔ 𝑤 ≈ suc 𝑥))
123	122	rspcev 3561	. . . . . . . . . . . . . . 15 ⊢ ((suc 𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ suc 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧)
124	121, 123	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧))
125	124	exlimdv 1936	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧))
126		df-rex 3070	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥))
127		breq2 5078	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑧))
128	127	cbvrexvw 3384	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥 ↔ ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧)
129	125, 126, 128	3imtr4g 296	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
130	129	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
131	90, 130	syld 47	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
132	131	expl 458	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑤 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
133		eleq1w 2821	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ ω ↔ 𝑦 ∈ ω))
134	133	pm5.32i 575	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑦 ∧ 𝑤 ∈ ω) ↔ (𝑤 = 𝑦 ∧ 𝑦 ∈ ω))
135	82	eqelsuc 6347	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑦 → 𝑤 ∈ suc 𝑦)
136		enrefnn 8837	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ω → 𝑤 ≈ 𝑤)
137		breq2 5078	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑤))
138	137	rspcev 3561	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑤) → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)
139	135, 136, 138	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝑤 = 𝑦 ∧ 𝑤 ∈ ω) → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)
140	139	2a1d 26	. . . . . . . . . . . . 13 ⊢ ((𝑤 = 𝑦 ∧ 𝑤 ∈ ω) → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
141	134, 140	sylbir 234	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑦 ∧ 𝑦 ∈ ω) → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
142	141	ex 413	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑦 ∈ ω → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
143	142	adantrd 492	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))))
144	143	pm2.43d 53	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
145	50, 132, 144	pm2.61ii 183	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
146	145	ex 413	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
147	24, 25, 146	alrimd 2208	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → ∀𝑤(𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
148	8, 12, 16, 20, 23, 147	finds 7745	. . . . 5 ⊢ (𝐴 ∈ ω → ∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))
149		psseq1 4022	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑤 ⊊ 𝐴 ↔ 𝐵 ⊊ 𝐴))
150		breq1 5077	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → (𝑤 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥))
151	150	rexbidv 3226	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))
152	149, 151	imbi12d 345	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥) ↔ (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
153	152	spcgv 3535	. . . . 5 ⊢ (𝐵 ∈ V → (∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥) → (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
154	148, 153	syl5 34	. . . 4 ⊢ (𝐵 ∈ V → (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
155	154	com3l 89	. . 3 ⊢ (𝐴 ∈ ω → (𝐵 ⊊ 𝐴 → (𝐵 ∈ V → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
156	155	imp 407	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (𝐵 ∈ V → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))
157	4, 156	mpd 15	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887   ⊊ wpss 3888  ∅c0 4256  {csn 4561  ⟨cop 4567   class class class wbr 5074  Ord word 6265  suc csuc 6268  –1-1-onto→wf1o 6432  ωcom 7712   ≈ cen 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-om 7713  df-en 8734

This theorem is referenced by:  ssnnfi  8952  ssnnfiOLD  8953