Theorem pssnn 8582

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.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pssnn

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

Step	Hyp	Ref	Expression
1		pssss 3993	. . . 4 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴)
2		ssexg 5118	. . . 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 3986	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ ∅))
6		rexeq 3366	. . . . . . . 8 ⊢ (𝑧 = ∅ → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥))
7	5, 6	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = ∅ → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)))
8	7	albidv 1898	. . . . . 6 ⊢ (𝑧 = ∅ → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)))
9		psseq2 3986	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝑦))
10		rexeq 3366	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥))
11	9, 10	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)))
12	11	albidv 1898	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)))
13		psseq2 3986	. . . . . . . 8 ⊢ (𝑧 = suc 𝑦 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ suc 𝑦))
14		rexeq 3366	. . . . . . . 8 ⊢ (𝑧 = suc 𝑦 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
15	13, 14	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = suc 𝑦 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
16	15	albidv 1898	. . . . . 6 ⊢ (𝑧 = suc 𝑦 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
17		psseq2 3986	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑤 ⊊ 𝑧 ↔ 𝑤 ⊊ 𝐴))
18		rexeq 3366	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))
19	17, 18	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ (𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)))
20	19	albidv 1898	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∀𝑤(𝑤 ⊊ 𝑧 → ∃𝑥 ∈ 𝑧 𝑤 ≈ 𝑥) ↔ ∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥)))
21		npss0 4311	. . . . . . . 8 ⊢ ¬ 𝑤 ⊊ ∅
22	21	pm2.21i 119	. . . . . . 7 ⊢ (𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)
23	22	ax-gen 1777	. . . . . 6 ⊢ ∀𝑤(𝑤 ⊊ ∅ → ∃𝑥 ∈ ∅ 𝑤 ≈ 𝑥)
24		nfv 1892	. . . . . . 7 ⊢ Ⅎ𝑤 𝑦 ∈ ω
25		nfa1 2121	. . . . . . 7 ⊢ Ⅎ𝑤∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)
26		elequ1 2088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑦 ∈ 𝑤))
27	26	biimpcd 250	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝑤 → (𝑧 = 𝑦 → 𝑦 ∈ 𝑤))
28	27	con3d 155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑤 → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦))
29	28	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ 𝑧 ∈ 𝑤) → (¬ 𝑦 ∈ 𝑤 → ¬ 𝑧 = 𝑦))
30		pssss 3993	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ⊊ suc 𝑦 → 𝑤 ⊆ suc 𝑦)
31	30	sseld 3888	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ⊊ suc 𝑦 → (𝑧 ∈ 𝑤 → 𝑧 ∈ suc 𝑦))
32		elsuci 6132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ suc 𝑦 → (𝑧 ∈ 𝑦 ∨ 𝑧 = 𝑦))
33	32	ord 859	. . . . . . . . . . . . . . . . . . . . . 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 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 3895	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) → 𝑤 ⊆ 𝑦)
40	39	anim1i 614	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦))
41		dfpss2 3983	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ⊊ 𝑦 ↔ (𝑤 ⊆ 𝑦 ∧ ¬ 𝑤 = 𝑦))
42	40, 41	sylibr 235	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ⊊ suc 𝑦 ∧ ¬ 𝑦 ∈ 𝑤) ∧ ¬ 𝑤 = 𝑦) → 𝑤 ⊊ 𝑦)
43		elelsuc 6138	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ∈ suc 𝑦)
44	43	anim1i 614	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑤 ≈ 𝑥) → (𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑥))
45	44	reximi2 3208	. . . . . . . . . . . . . 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 2148	. . . . . . . . . . 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 4037	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ⊆ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦}))
53		nnord 7444	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → Ord 𝑦)
54		orddif 6159	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑦 → 𝑦 = (suc 𝑦 ∖ {𝑦}))
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ω → 𝑦 = (suc 𝑦 ∖ {𝑦}))
56	55	sseq2d 3920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ω → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊆ (suc 𝑦 ∖ {𝑦})))
57	52, 56	syl5ibr 247	. . . . . . . . . . . . . . . 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 4334	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ⊊ suc 𝑦 → ∃𝑧(𝑧 ∈ suc 𝑦 ∧ ¬ 𝑧 ∈ 𝑤))
60		eleq2 2871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∖ {𝑦}) = 𝑦 → (𝑧 ∈ (𝑤 ∖ {𝑦}) ↔ 𝑧 ∈ 𝑦))
61		eldifi 4024	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝑤 ∖ {𝑦}) → 𝑧 ∈ 𝑤)
62	60, 61	syl6bir 255	. . . . . . . . . . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . . . . . . . . . . . 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 180	. . . . . . . . . . . . . . . . . . . 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 155	. . . . . . . . . . . . . . . . . 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 1911	. . . . . . . . . . . . . . . 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 620	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑤 ⊊ suc 𝑦 ∧ 𝑤 ⊊ suc 𝑦) → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦)))
74	51, 73	syl5bir 244	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦)))
75		dfpss2 3983	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 ↔ ((𝑤 ∖ {𝑦}) ⊆ 𝑦 ∧ ¬ (𝑤 ∖ {𝑦}) = 𝑦))
76	74, 75	syl6ibr 253	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (𝑤 ⊊ suc 𝑦 → (𝑤 ∖ {𝑦}) ⊊ 𝑦))
77		psseq1 3985	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (𝑤 ⊊ 𝑦 ↔ 𝑧 ⊊ 𝑦))
78		breq1 4965	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑧 ≈ 𝑥))
79	78	rexbidv 3260	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑧 → (∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥))
80	77, 79	imbi12d 346	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → ((𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ (𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥)))
81	80	cbvalvw 2020	. . . . . . . . . . . . 13 ⊢ (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) ↔ ∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥))
82		vex 3440	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
83	82	difexi 5123	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∖ {𝑦}) ∈ V
84		psseq1 3985	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ⊊ 𝑦 ↔ (𝑤 ∖ {𝑦}) ⊊ 𝑦))
85		breq1 4965	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (𝑧 ≈ 𝑥 ↔ (𝑤 ∖ {𝑦}) ≈ 𝑥))
86	85	rexbidv 3260	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → (∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
87	84, 86	imbi12d 346	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑤 ∖ {𝑦}) → ((𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) ↔ ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥)))
88	83, 87	spcv 3548	. . . . . . . . . . . . 13 ⊢ (∀𝑧(𝑧 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑧 ≈ 𝑥) → ((𝑤 ∖ {𝑦}) ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥))
89	81, 88	sylbi 218	. . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Ord 𝑦 → (𝑥 ∈ 𝑦 ↔ suc 𝑥 ∈ suc 𝑦))
92	91	biimpd 230	. . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω) → 𝑥 ∈ ω)
97		snex 5223	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑦, 𝑥⟩} ∈ V
98		vex 3440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑦 ∈ V
99		vex 3440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑥 ∈ V
100	98, 99	f1osn 6522	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑦, 𝑥⟩}:{𝑦}–1-1-onto→{𝑥}
101		f1oen3g 8373	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({⟨𝑦, 𝑥⟩} ∈ V ∧ {⟨𝑦, 𝑥⟩}:{𝑦}–1-1-onto→{𝑥}) → {𝑦} ≈ {𝑥})
102	97, 100, 101	mp2an 688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑦} ≈ {𝑥}
103	102	jctr 525	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∖ {𝑦}) ≈ 𝑥 → ((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥}))
104		nnord 7444	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ω → Ord 𝑥)
105		orddisj 6104	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
106	104, 105	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
107		incom 4099	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝑦} ∩ (𝑤 ∖ {𝑦})) = ((𝑤 ∖ {𝑦}) ∩ {𝑦})
108		disjdif 4335	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({𝑦} ∩ (𝑤 ∖ {𝑦})) = ∅
109	107, 108	eqtr3i 2821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅
110	106, 109	jctil 520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ω → (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅))
111		unen 8444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ {𝑦} ≈ {𝑥}) ∧ (((𝑤 ∖ {𝑦}) ∩ {𝑦}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥}))
112	103, 110, 111	syl2an 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥}))
113		difsnid 4650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝑤 → ((𝑤 ∖ {𝑦}) ∪ {𝑦}) = 𝑤)
114	113	eqcomd 2801	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑤 → 𝑤 = ((𝑤 ∖ {𝑦}) ∪ {𝑦}))
115		df-suc 6072	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
116	115	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑤 → suc 𝑥 = (𝑥 ∪ {𝑥}))
117	114, 116	breq12d 4975	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝑤 → (𝑤 ≈ suc 𝑥 ↔ ((𝑤 ∖ {𝑦}) ∪ {𝑦}) ≈ (𝑥 ∪ {𝑥})))
118	112, 117	syl5ibr 247	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑤 ≈ suc 𝑥))
119	96, 118	sylan2i 605	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑤 → (((𝑤 ∖ {𝑦}) ≈ 𝑥 ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ω)) → 𝑤 ≈ suc 𝑥))
120	119	exp4d 434	. . . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → 𝑤 ≈ suc 𝑥))
123	95, 122	jcad 513	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → ((𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → (suc 𝑥 ∈ suc 𝑦 ∧ 𝑤 ≈ suc 𝑥)))
124		breq2 4966	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc 𝑥 → (𝑤 ≈ 𝑧 ↔ 𝑤 ≈ suc 𝑥))
125	124	rspcev 3559	. . . . . . . . . . . . . . 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 1911	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥) → ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧))
128		df-rex 3111	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ (𝑤 ∖ {𝑦}) ≈ 𝑥))
129		breq2 4966	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑧))
130	129	cbvrexv 3404	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥 ↔ ∃𝑧 ∈ suc 𝑦𝑤 ≈ 𝑧)
131	127, 128, 130	3imtr4g 297	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ω) → (∃𝑥 ∈ 𝑦 (𝑤 ∖ {𝑦}) ≈ 𝑥 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
132	131	adantr 481	. . . . . . . . . . 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 458	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑤 → ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
135	82	eqelsuc 6147	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → 𝑤 ∈ suc 𝑦)
136	82	enref 8390	. . . . . . . . . . 11 ⊢ 𝑤 ≈ 𝑤
137		breq2 4966	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (𝑤 ≈ 𝑥 ↔ 𝑤 ≈ 𝑤))
138	137	rspcev 3559	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ suc 𝑦 ∧ 𝑤 ≈ 𝑤) → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)
139	135, 136, 138	sylancl 586	. . . . . . . . . 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 184	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ ∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥)) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥))
142	141	ex 413	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑤(𝑤 ⊊ 𝑦 → ∃𝑥 ∈ 𝑦 𝑤 ≈ 𝑥) → (𝑤 ⊊ suc 𝑦 → ∃𝑥 ∈ suc 𝑦𝑤 ≈ 𝑥)))
143	24, 25, 142	alrimd 2180	. . . . . 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 7464	. . . . 5 ⊢ (𝐴 ∈ ω → ∀𝑤(𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥))
145		psseq1 3985	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (𝑤 ⊊ 𝐴 ↔ 𝐵 ⊊ 𝐴))
146		breq1 4965	. . . . . . . 8 ⊢ (𝑤 = 𝐵 → (𝑤 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥))
147	146	rexbidv 3260	. . . . . . 7 ⊢ (𝑤 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥))
148	145, 147	imbi12d 346	. . . . . 6 ⊢ (𝑤 = 𝐵 → ((𝑤 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝑤 ≈ 𝑥) ↔ (𝐵 ⊊ 𝐴 → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)))
149	148	spcgv 3538	. . . . 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 407	. 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 396  ∀wal 1520   = wceq 1522  ∃wex 1761   ∈ wcel 2081  ∃wrex 3106  Vcvv 3437   ∖ cdif 3856   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859   ⊊ wpss 3860  ∅c0 4211  {csn 4472  ⟨cop 4478   class class class wbr 4962  Ord word 6065  suc csuc 6068  –1-1-onto→wf1o 6224  ωcom 7436   ≈ cen 8354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-om 7437  df-en 8358

This theorem is referenced by:  ssnnfi  8583