Theorem inf3lem2 9538

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9544 for detailed description. (Contributed by NM, 28-Oct-1996.)

Hypotheses

Ref	Expression
inf3lem.1	⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
inf3lem.2	⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3	⊢ 𝐴 ∈ V
inf3lem.4	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
inf3lem2	⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ 𝑥))

Distinct variable group:   𝑥,𝑦,𝑤

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑤)   𝐵(𝑥,𝑦,𝑤)   𝐹(𝑥,𝑦,𝑤)   𝐺(𝑥,𝑦,𝑤)

Proof of Theorem inf3lem2

Dummy variables 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . . 5 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2	1	neeq1d 2991	. . . 4 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘∅) ≠ 𝑥))
3	2	imbi2d 340	. . 3 ⊢ (𝑣 = ∅ → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑣) ≠ 𝑥) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘∅) ≠ 𝑥)))
4		fveq2 6834	. . . . 5 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
5	4	neeq1d 2991	. . . 4 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘𝑢) ≠ 𝑥))
6	5	imbi2d 340	. . 3 ⊢ (𝑣 = 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑣) ≠ 𝑥) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑢) ≠ 𝑥)))
7		fveq2 6834	. . . . 5 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
8	7	neeq1d 2991	. . . 4 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘suc 𝑢) ≠ 𝑥))
9	8	imbi2d 340	. . 3 ⊢ (𝑣 = suc 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑣) ≠ 𝑥) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥)))
10		fveq2 6834	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
11	10	neeq1d 2991	. . . 4 ⊢ (𝑣 = 𝐴 → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘𝐴) ≠ 𝑥))
12	11	imbi2d 340	. . 3 ⊢ (𝑣 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑣) ≠ 𝑥) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐴) ≠ 𝑥)))
13		inf3lem.1	. . . . . . . 8 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
14		inf3lem.2	. . . . . . . 8 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
15		inf3lem.3	. . . . . . . 8 ⊢ 𝐴 ∈ V
16		inf3lem.4	. . . . . . . 8 ⊢ 𝐵 ∈ V
17	13, 14, 15, 16	inf3lemb 9534	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
18	17	eqeq1i 2741	. . . . . 6 ⊢ ((𝐹‘∅) = 𝑥 ↔ ∅ = 𝑥)
19		eqcom 2743	. . . . . 6 ⊢ (∅ = 𝑥 ↔ 𝑥 = ∅)
20	18, 19	sylbb 219	. . . . 5 ⊢ ((𝐹‘∅) = 𝑥 → 𝑥 = ∅)
21	20	necon3i 2964	. . . 4 ⊢ (𝑥 ≠ ∅ → (𝐹‘∅) ≠ 𝑥)
22	21	adantr 480	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘∅) ≠ 𝑥)
23		vex 3444	. . . . . . . . 9 ⊢ 𝑢 ∈ V
24	13, 14, 23, 16	inf3lemd 9536	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝐹‘𝑢) ⊆ 𝑥)
25		df-pss 3921	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢) ⊊ 𝑥 ↔ ((𝐹‘𝑢) ⊆ 𝑥 ∧ (𝐹‘𝑢) ≠ 𝑥))
26		pssnel 4423	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢) ⊊ 𝑥 → ∃𝑣(𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)))
27	25, 26	sylbir 235	. . . . . . . . 9 ⊢ (((𝐹‘𝑢) ⊆ 𝑥 ∧ (𝐹‘𝑢) ≠ 𝑥) → ∃𝑣(𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)))
28		ssel 3927	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ ∪ 𝑥 → (𝑣 ∈ 𝑥 → 𝑣 ∈ ∪ 𝑥))
29		eluni 4866	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ∪ 𝑥 ↔ ∃𝑓(𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥))
30	28, 29	imbitrdi 251	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ∪ 𝑥 → (𝑣 ∈ 𝑥 → ∃𝑓(𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥)))
31		eleq2 2825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘suc 𝑢) = 𝑥 → (𝑓 ∈ (𝐹‘suc 𝑢) ↔ 𝑓 ∈ 𝑥))
32	31	biimparc 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ 𝑥 ∧ (𝐹‘suc 𝑢) = 𝑥) → 𝑓 ∈ (𝐹‘suc 𝑢))
33	13, 14, 23, 16	inf3lemc 9535	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
34	33	eleq2d 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ∈ ω → (𝑓 ∈ (𝐹‘suc 𝑢) ↔ 𝑓 ∈ (𝐺‘(𝐹‘𝑢))))
35		elin 3917	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ (𝑓 ∩ 𝑥) ↔ (𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥))
36		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑓 ∈ V
37		fvex 6847	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹‘𝑢) ∈ V
38	13, 14, 36, 37	inf3lema 9533	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) ↔ (𝑓 ∈ 𝑥 ∧ (𝑓 ∩ 𝑥) ⊆ (𝐹‘𝑢)))
39	38	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) → (𝑓 ∩ 𝑥) ⊆ (𝐹‘𝑢))
40	39	sseld 3932	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) → (𝑣 ∈ (𝑓 ∩ 𝑥) → 𝑣 ∈ (𝐹‘𝑢)))
41	35, 40	biimtrrid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ (𝐹‘𝑢)))
42	34, 41	biimtrdi 253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ ω → (𝑓 ∈ (𝐹‘suc 𝑢) → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ (𝐹‘𝑢))))
43	32, 42	syl5 34	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ω → ((𝑓 ∈ 𝑥 ∧ (𝐹‘suc 𝑢) = 𝑥) → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ (𝐹‘𝑢))))
44	43	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ ω → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → ((𝑓 ∈ 𝑥 ∧ (𝐹‘suc 𝑢) = 𝑥) → 𝑣 ∈ (𝐹‘𝑢))))
45	44	exp5c 444	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ ω → (𝑣 ∈ 𝑓 → (𝑣 ∈ 𝑥 → (𝑓 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢))))))
46	45	com34 91	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ω → (𝑣 ∈ 𝑓 → (𝑓 ∈ 𝑥 → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢))))))
47	46	impd 410	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ω → ((𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥) → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))))
48	47	exlimdv 1934	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ω → (∃𝑓(𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥) → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))))
49	30, 48	sylan9r 508	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑣 ∈ 𝑥 → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))))
50	49	pm2.43d 53	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢))))
51		id 22	. . . . . . . . . . . . 13 ⊢ (((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)) → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))
52	51	necon3bd 2946	. . . . . . . . . . . 12 ⊢ (((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)) → (¬ 𝑣 ∈ (𝐹‘𝑢) → (𝐹‘suc 𝑢) ≠ 𝑥))
53	50, 52	syl6 35	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑣 ∈ 𝑥 → (¬ 𝑣 ∈ (𝐹‘𝑢) → (𝐹‘suc 𝑢) ≠ 𝑥)))
54	53	impd 410	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)) → (𝐹‘suc 𝑢) ≠ 𝑥))
55	54	exlimdv 1934	. . . . . . . . 9 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (∃𝑣(𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)) → (𝐹‘suc 𝑢) ≠ 𝑥))
56	27, 55	syl5 34	. . . . . . . 8 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (((𝐹‘𝑢) ⊆ 𝑥 ∧ (𝐹‘𝑢) ≠ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥))
57	24, 56	sylani 604	. . . . . . 7 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ≠ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥))
58	57	exp4b 430	. . . . . 6 ⊢ (𝑢 ∈ ω → (𝑥 ⊆ ∪ 𝑥 → (𝑢 ∈ ω → ((𝐹‘𝑢) ≠ 𝑥 → (𝐹‘suc 𝑢) ≠ 𝑥))))
59	58	pm2.43a 54	. . . . 5 ⊢ (𝑢 ∈ ω → (𝑥 ⊆ ∪ 𝑥 → ((𝐹‘𝑢) ≠ 𝑥 → (𝐹‘suc 𝑢) ≠ 𝑥)))
60	59	adantld 490	. . . 4 ⊢ (𝑢 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐹‘𝑢) ≠ 𝑥 → (𝐹‘suc 𝑢) ≠ 𝑥)))
61	60	a2d 29	. . 3 ⊢ (𝑢 ∈ ω → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑢) ≠ 𝑥) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥)))
62	3, 6, 9, 12, 22, 61	finds 7838	. 2 ⊢ (𝐴 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐴) ≠ 𝑥))
63	62	com12 32	1 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   ⊊ wpss 3902  ∅c0 4285  ∪ cuni 4863   ↦ cmpt 5179   ↾ cres 5626  suc csuc 6319  ‘cfv 6492  ωcom 7808  reccrdg 8340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  inf3lem3  9539