Theorem inf3lem2 9623

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9629 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 6883	. . . . 5 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2	1	neeq1d 3015	. . . 4 ⊢ (𝑣 = ∅ → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘∅) ≠ 𝑥))
3	2	imbi2d 343	. . 3 ⊢ (𝑣 = ∅ → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑣) ≠ 𝑥) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘∅) ≠ 𝑥)))
4		fveq2 6883	. . . . 5 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
5	4	neeq1d 3015	. . . 4 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘𝑢) ≠ 𝑥))
6	5	imbi2d 343	. . 3 ⊢ (𝑣 = 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑣) ≠ 𝑥) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑢) ≠ 𝑥)))
7		fveq2 6883	. . . . 5 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
8	7	neeq1d 3015	. . . 4 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘suc 𝑢) ≠ 𝑥))
9	8	imbi2d 343	. . 3 ⊢ (𝑣 = suc 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑣) ≠ 𝑥) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥)))
10		fveq2 6883	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
11	10	neeq1d 3015	. . . 4 ⊢ (𝑣 = 𝐴 → ((𝐹‘𝑣) ≠ 𝑥 ↔ (𝐹‘𝐴) ≠ 𝑥))
12	11	imbi2d 343	. . 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 9619	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
18	17	eqeq1i 2766	. . . . . 6 ⊢ ((𝐹‘∅) = 𝑥 ↔ ∅ = 𝑥)
19		eqcom 2768	. . . . . 6 ⊢ (∅ = 𝑥 ↔ 𝑥 = ∅)
20	18, 19	sylbb 222	. . . . 5 ⊢ ((𝐹‘∅) = 𝑥 → 𝑥 = ∅)
21	20	necon3i 2988	. . . 4 ⊢ (𝑥 ≠ ∅ → (𝐹‘∅) ≠ 𝑥)
22	21	adantr 486	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘∅) ≠ 𝑥)
23		vex 3455	. . . . . . . . 9 ⊢ 𝑢 ∈ V
24	13, 14, 23, 16	inf3lemd 9621	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝐹‘𝑢) ⊆ 𝑥)
25		df-pss 3919	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢) ⊊ 𝑥 ↔ ((𝐹‘𝑢) ⊆ 𝑥 ∧ (𝐹‘𝑢) ≠ 𝑥))
26		pssnel 4424	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢) ⊊ 𝑥 → ∃𝑣(𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)))
27	25, 26	sylbir 238	. . . . . . . . 9 ⊢ (((𝐹‘𝑢) ⊆ 𝑥 ∧ (𝐹‘𝑢) ≠ 𝑥) → ∃𝑣(𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)))
28		ssel 3925	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ ∪ 𝑥 → (𝑣 ∈ 𝑥 → 𝑣 ∈ ∪ 𝑥))
29		eluni 4870	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ ∪ 𝑥 ↔ ∃𝑓(𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥))
30	28, 29	imbitrdi 254	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ∪ 𝑥 → (𝑣 ∈ 𝑥 → ∃𝑓(𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥)))
31		eleq2 2850	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘suc 𝑢) = 𝑥 → (𝑓 ∈ (𝐹‘suc 𝑢) ↔ 𝑓 ∈ 𝑥))
32	31	biimparc 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ 𝑥 ∧ (𝐹‘suc 𝑢) = 𝑥) → 𝑓 ∈ (𝐹‘suc 𝑢))
33	13, 14, 23, 16	inf3lemc 9620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
34	33	eleq2d 2847	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ∈ ω → (𝑓 ∈ (𝐹‘suc 𝑢) ↔ 𝑓 ∈ (𝐺‘(𝐹‘𝑢))))
35		elin 3915	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ (𝑓 ∩ 𝑥) ↔ (𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥))
36		vex 3455	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑓 ∈ V
37		fvex 6896	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹‘𝑢) ∈ V
38	13, 14, 36, 37	inf3lema 9618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) ↔ (𝑓 ∈ 𝑥 ∧ (𝑓 ∩ 𝑥) ⊆ (𝐹‘𝑢)))
39	38	simprbi 503	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) → (𝑓 ∩ 𝑥) ⊆ (𝐹‘𝑢))
40	39	sseld 3930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) → (𝑣 ∈ (𝑓 ∩ 𝑥) → 𝑣 ∈ (𝐹‘𝑢)))
41	35, 40	biimtrrid 246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 ∈ (𝐺‘(𝐹‘𝑢)) → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ (𝐹‘𝑢)))
42	34, 41	biimtrdi 256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ ω → (𝑓 ∈ (𝐹‘suc 𝑢) → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ (𝐹‘𝑢))))
43	32, 42	syl5 35	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ω → ((𝑓 ∈ 𝑥 ∧ (𝐹‘suc 𝑢) = 𝑥) → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → 𝑣 ∈ (𝐹‘𝑢))))
44	43	com23 87	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ ω → ((𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥) → ((𝑓 ∈ 𝑥 ∧ (𝐹‘suc 𝑢) = 𝑥) → 𝑣 ∈ (𝐹‘𝑢))))
45	44	exp5c 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ ω → (𝑣 ∈ 𝑓 → (𝑣 ∈ 𝑥 → (𝑓 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢))))))
46	45	com34 92	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ ω → (𝑣 ∈ 𝑓 → (𝑓 ∈ 𝑥 → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢))))))
47	46	impd 416	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ω → ((𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥) → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))))
48	47	exlimdv 1966	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ω → (∃𝑓(𝑣 ∈ 𝑓 ∧ 𝑓 ∈ 𝑥) → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))))
49	30, 48	sylan9r 518	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑣 ∈ 𝑥 → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))))
50	49	pm2.43d 54	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑣 ∈ 𝑥 → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢))))
51		id 23	. . . . . . . . . . . . 13 ⊢ (((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)) → ((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)))
52	51	necon3bd 2970	. . . . . . . . . . . 12 ⊢ (((𝐹‘suc 𝑢) = 𝑥 → 𝑣 ∈ (𝐹‘𝑢)) → (¬ 𝑣 ∈ (𝐹‘𝑢) → (𝐹‘suc 𝑢) ≠ 𝑥))
53	50, 52	syl6 36	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑣 ∈ 𝑥 → (¬ 𝑣 ∈ (𝐹‘𝑢) → (𝐹‘suc 𝑢) ≠ 𝑥)))
54	53	impd 416	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)) → (𝐹‘suc 𝑢) ≠ 𝑥))
55	54	exlimdv 1966	. . . . . . . . 9 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (∃𝑣(𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)) → (𝐹‘suc 𝑢) ≠ 𝑥))
56	27, 55	syl5 35	. . . . . . . 8 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → (((𝐹‘𝑢) ⊆ 𝑥 ∧ (𝐹‘𝑢) ≠ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥))
57	24, 56	sylani 616	. . . . . . 7 ⊢ ((𝑢 ∈ ω ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑢 ∈ ω ∧ (𝐹‘𝑢) ≠ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥))
58	57	exp4b 436	. . . . . 6 ⊢ (𝑢 ∈ ω → (𝑥 ⊆ ∪ 𝑥 → (𝑢 ∈ ω → ((𝐹‘𝑢) ≠ 𝑥 → (𝐹‘suc 𝑢) ≠ 𝑥))))
59	58	pm2.43a 55	. . . . 5 ⊢ (𝑢 ∈ ω → (𝑥 ⊆ ∪ 𝑥 → ((𝐹‘𝑢) ≠ 𝑥 → (𝐹‘suc 𝑢) ≠ 𝑥)))
60	59	adantld 496	. . . 4 ⊢ (𝑢 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐹‘𝑢) ≠ 𝑥 → (𝐹‘suc 𝑢) ≠ 𝑥)))
61	60	a2d 30	. . 3 ⊢ (𝑢 ∈ ω → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝑢) ≠ 𝑥) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘suc 𝑢) ≠ 𝑥)))
62	3, 6, 9, 12, 22, 61	finds 7906	. 2 ⊢ (𝐴 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐴) ≠ 𝑥))
63	62	com12 33	1 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  {crab 3413  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899   ⊊ wpss 3900  ∅c0 4279  ∪ cuni 4867   ↦ cmpt 5186   ↾ cres 5653  suc csuc 6363  ‘cfv 6537  ωcom 7875  reccrdg 8410

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411

This theorem is used by:  inf3lem3  9624