Theorem inf3lem2 9544

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9550 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 6826	. . . . 5 ⊢ (𝑣 = ∅ → (𝐹‘𝑣) = (𝐹‘∅))
2	1	neeq1d 2984	. . . 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 6826	. . . . 5 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
5	4	neeq1d 2984	. . . 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 6826	. . . . 5 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
8	7	neeq1d 2984	. . . 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 6826	. . . . 5 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
11	10	neeq1d 2984	. . . 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 9540	. . . . . . 7 ⊢ (𝐹‘∅) = ∅
18	17	eqeq1i 2734	. . . . . 6 ⊢ ((𝐹‘∅) = 𝑥 ↔ ∅ = 𝑥)
19		eqcom 2736	. . . . . 6 ⊢ (∅ = 𝑥 ↔ 𝑥 = ∅)
20	18, 19	sylbb 219	. . . . 5 ⊢ ((𝐹‘∅) = 𝑥 → 𝑥 = ∅)
21	20	necon3i 2957	. . . 4 ⊢ (𝑥 ≠ ∅ → (𝐹‘∅) ≠ 𝑥)
22	21	adantr 480	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘∅) ≠ 𝑥)
23		vex 3442	. . . . . . . . 9 ⊢ 𝑢 ∈ V
24	13, 14, 23, 16	inf3lemd 9542	. . . . . . . 8 ⊢ (𝑢 ∈ ω → (𝐹‘𝑢) ⊆ 𝑥)
25		df-pss 3925	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (((𝐹‘𝑢) ⊆ 𝑥 ∧ (𝐹‘𝑢) ≠ 𝑥) → ∃𝑣(𝑣 ∈ 𝑥 ∧ ¬ 𝑣 ∈ (𝐹‘𝑢)))
28		ssel 3931	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ ∪ 𝑥 → (𝑣 ∈ 𝑥 → 𝑣 ∈ ∪ 𝑥))
29		eluni 4864	. . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . 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 9541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 ∈ ω → (𝐹‘suc 𝑢) = (𝐺‘(𝐹‘𝑢)))
34	33	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ∈ ω → (𝑓 ∈ (𝐹‘suc 𝑢) ↔ 𝑓 ∈ (𝐺‘(𝐹‘𝑢))))
35		elin 3921	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ (𝑓 ∩ 𝑥) ↔ (𝑣 ∈ 𝑓 ∧ 𝑣 ∈ 𝑥))
36		vex 3442	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑓 ∈ V
37		fvex 6839	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐹‘𝑢) ∈ V
38	13, 14, 36, 37	inf3lema 9539	. . . . . . . . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . . . . . . 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 1933	. . . . . . . . . . . . . 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 2939	. . . . . . . . . . . 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 1933	. . . . . . . . 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 7836	. 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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  {crab 3396  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905   ⊊ wpss 3906  ∅c0 4286  ∪ cuni 4861   ↦ cmpt 5176   ↾ cres 5625  suc csuc 6313  ‘cfv 6486  ωcom 7806  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  inf3lem3  9545