Theorem inf3lemd 3463

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 3471 for detailed description.

Hypotheses

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

Assertion

Ref	Expression
inf3lemd	⊢ (A ∈ ω → (F ‘A) ⊆ x)

Distinct variable group(s):   x,y,z,w

Proof of Theorem inf3lemd

Step	Hyp	Ref	Expression
1		nnsuc 2389	. . . 4 ⊢ ((A ∈ ω ∧ ¬ A = ∅) → ∃v ∈ ω A = suc v)
2		fveq2 2832	. . . . . . . 8 ⊢ (A = suc v → (F ‘A) = (F ‘suc v))
3	2	sseq1d 1527	. . . . . . 7 ⊢ (A = suc v → ((F ‘A) ⊆ x ↔ (F ‘suc v) ⊆ x))
4		inf3lem.1	. . . . . . . . . . 11 ⊢ G = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
5		inf3lem.2	. . . . . . . . . . 11 ⊢ F = (rec(G, ∅) ↾ ω)
6		visset 1350	. . . . . . . . . . 11 ⊢ v ∈ V
7		inf3lem.4	. . . . . . . . . . 11 ⊢ B ∈ V
8	4, 5, 6, 7	inf3lemc 3462	. . . . . . . . . 10 ⊢ (v ∈ ω → (F ‘suc v) = (G ‘(F ‘v)))
9	8	eleq2d 1156	. . . . . . . . 9 ⊢ (v ∈ ω → (u ∈ (F ‘suc v) ↔ u ∈ (G ‘(F ‘v))))
10		visset 1350	. . . . . . . . . . 11 ⊢ u ∈ V
11		fvex 2838	. . . . . . . . . . 11 ⊢ (F ‘v) ∈ V
12	4, 5, 10, 11	inf3lema 3460	. . . . . . . . . 10 ⊢ (u ∈ (G ‘(F ‘v)) ↔ (u ∈ x ∧ (u ∩ x) ⊆ (F ‘v)))
13	12	pm3.26bd 259	. . . . . . . . 9 ⊢ (u ∈ (G ‘(F ‘v)) → u ∈ x)
14	9, 13	syl6bi 187	. . . . . . . 8 ⊢ (v ∈ ω → (u ∈ (F ‘suc v) → u ∈ x))
15	14	ssrdv 1509	. . . . . . 7 ⊢ (v ∈ ω → (F ‘suc v) ⊆ x)
16	3, 15	syl5bir 184	. . . . . 6 ⊢ (A = suc v → (v ∈ ω → (F ‘A) ⊆ x))
17	16	com12 13	. . . . 5 ⊢ (v ∈ ω → (A = suc v → (F ‘A) ⊆ x))
18	17	r19.23aiv 1284	. . . 4 ⊢ (∃v ∈ ω A = suc v → (F ‘A) ⊆ x)
19	1, 18	syl 12	. . 3 ⊢ ((A ∈ ω ∧ ¬ A = ∅) → (F ‘A) ⊆ x)
20	19	exp 291	. 2 ⊢ (A ∈ ω → (¬ A = ∅ → (F ‘A) ⊆ x))
21		0ss 1725	. . 3 ⊢ ∅ ⊆ x
22		fveq2 2832	. . . . 5 ⊢ (A = ∅ → (F ‘A) = (F ‘∅))
23		inf3lem.3	. . . . . 6 ⊢ A ∈ V
24	4, 5, 23, 7	inf3lemb 3461	. . . . 5 ⊢ (F ‘∅) = ∅
25	22, 24	syl6eq 1140	. . . 4 ⊢ (A = ∅ → (F ‘A) = ∅)
26	25	sseq1d 1527	. . 3 ⊢ (A = ∅ → ((F ‘A) ⊆ x ↔ ∅ ⊆ x))
27	21, 26	mpbiri 169	. 2 ⊢ (A = ∅ → (F ‘A) ⊆ x)
28	20, 27	pm2.61d2 111	1 ⊢ (A ∈ ω → (F ‘A) ⊆ x)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {copab 2055  suc csuc 2201  ωcom 2372   ↾ cres 2412   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  inf3lem2 3465  inf3lem3 3466  inf3lem6 3469

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970

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