Theorem inf3lem1 3464

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
inf3lem1	⊢ (A ∈ ω → (F ‘A) ⊆ (F ‘suc A))

Distinct variable group(s):   x,y,z,w

Proof of Theorem inf3lem1

Step	Hyp	Ref	Expression
1		fveq2 2832	. . 3 ⊢ (v = ∅ → (F ‘v) = (F ‘∅))
2		suceq 2288	. . . 4 ⊢ (v = ∅ → suc v = suc ∅)
3	2	fveq2d 2836	. . 3 ⊢ (v = ∅ → (F ‘suc v) = (F ‘suc ∅))
4	1, 3	sseq12d 1529	. 2 ⊢ (v = ∅ → ((F ‘v) ⊆ (F ‘suc v) ↔ (F ‘∅) ⊆ (F ‘suc ∅)))
5		fveq2 2832	. . 3 ⊢ (v = u → (F ‘v) = (F ‘u))
6		suceq 2288	. . . 4 ⊢ (v = u → suc v = suc u)
7	6	fveq2d 2836	. . 3 ⊢ (v = u → (F ‘suc v) = (F ‘suc u))
8	5, 7	sseq12d 1529	. 2 ⊢ (v = u → ((F ‘v) ⊆ (F ‘suc v) ↔ (F ‘u) ⊆ (F ‘suc u)))
9		fveq2 2832	. . 3 ⊢ (v = suc u → (F ‘v) = (F ‘suc u))
10		suceq 2288	. . . 4 ⊢ (v = suc u → suc v = suc suc u)
11	10	fveq2d 2836	. . 3 ⊢ (v = suc u → (F ‘suc v) = (F ‘suc suc u))
12	9, 11	sseq12d 1529	. 2 ⊢ (v = suc u → ((F ‘v) ⊆ (F ‘suc v) ↔ (F ‘suc u) ⊆ (F ‘suc suc u)))
13		fveq2 2832	. . 3 ⊢ (v = A → (F ‘v) = (F ‘A))
14		suceq 2288	. . . 4 ⊢ (v = A → suc v = suc A)
15	14	fveq2d 2836	. . 3 ⊢ (v = A → (F ‘suc v) = (F ‘suc A))
16	13, 15	sseq12d 1529	. 2 ⊢ (v = A → ((F ‘v) ⊆ (F ‘suc v) ↔ (F ‘A) ⊆ (F ‘suc A)))
17		inf3lem.1	. . . 4 ⊢ G = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
18		inf3lem.2	. . . 4 ⊢ F = (rec(G, ∅) ↾ ω)
19		inf3lem.3	. . . 4 ⊢ A ∈ V
20	17, 18, 19, 19	inf3lemb 3461	. . 3 ⊢ (F ‘∅) = ∅
21		0ss 1725	. . 3 ⊢ ∅ ⊆ (F ‘suc ∅)
22	20, 21	eqsstr 1530	. 2 ⊢ (F ‘∅) ⊆ (F ‘suc ∅)
23		visset 1350	. . . . . . . . . 10 ⊢ u ∈ V
24	17, 18, 23, 23	inf3lemc 3462	. . . . . . . . 9 ⊢ (u ∈ ω → (F ‘suc u) = (G ‘(F ‘u)))
25	24	eleq2d 1156	. . . . . . . 8 ⊢ (u ∈ ω → (v ∈ (F ‘suc u) ↔ v ∈ (G ‘(F ‘u))))
26		visset 1350	. . . . . . . . 9 ⊢ v ∈ V
27		fvex 2838	. . . . . . . . 9 ⊢ (F ‘u) ∈ V
28	17, 18, 26, 27	inf3lema 3460	. . . . . . . 8 ⊢ (v ∈ (G ‘(F ‘u)) ↔ (v ∈ x ∧ (v ∩ x) ⊆ (F ‘u)))
29	25, 28	syl6bb 414	. . . . . . 7 ⊢ (u ∈ ω → (v ∈ (F ‘suc u) ↔ (v ∈ x ∧ (v ∩ x) ⊆ (F ‘u))))
30		peano2b 2388	. . . . . . . . . 10 ⊢ (u ∈ ω ↔ suc u ∈ ω)
31	23	sucex 2303	. . . . . . . . . . 11 ⊢ suc u ∈ V
32	17, 18, 31, 23	inf3lemc 3462	. . . . . . . . . 10 ⊢ (suc u ∈ ω → (F ‘suc suc u) = (G ‘(F ‘suc u)))
33	30, 32	sylbi 174	. . . . . . . . 9 ⊢ (u ∈ ω → (F ‘suc suc u) = (G ‘(F ‘suc u)))
34	33	eleq2d 1156	. . . . . . . 8 ⊢ (u ∈ ω → (v ∈ (F ‘suc suc u) ↔ v ∈ (G ‘(F ‘suc u))))
35		fvex 2838	. . . . . . . . 9 ⊢ (F ‘suc u) ∈ V
36	17, 18, 26, 35	inf3lema 3460	. . . . . . . 8 ⊢ (v ∈ (G ‘(F ‘suc u)) ↔ (v ∈ x ∧ (v ∩ x) ⊆ (F ‘suc u)))
37	34, 36	syl6bb 414	. . . . . . 7 ⊢ (u ∈ ω → (v ∈ (F ‘suc suc u) ↔ (v ∈ x ∧ (v ∩ x) ⊆ (F ‘suc u))))
38	29, 37	imbi12d 474	. . . . . 6 ⊢ (u ∈ ω → ((v ∈ (F ‘suc u) → v ∈ (F ‘suc suc u)) ↔ ((v ∈ x ∧ (v ∩ x) ⊆ (F ‘u)) → (v ∈ x ∧ (v ∩ x) ⊆ (F ‘suc u)))))
39		sstr2 1510	. . . . . . . 8 ⊢ ((v ∩ x) ⊆ (F ‘u) → ((F ‘u) ⊆ (F ‘suc u) → (v ∩ x) ⊆ (F ‘suc u)))
40	39	com12 13	. . . . . . 7 ⊢ ((F ‘u) ⊆ (F ‘suc u) → ((v ∩ x) ⊆ (F ‘u) → (v ∩ x) ⊆ (F ‘suc u)))
41	40	anim2d 433	. . . . . 6 ⊢ ((F ‘u) ⊆ (F ‘suc u) → ((v ∈ x ∧ (v ∩ x) ⊆ (F ‘u)) → (v ∈ x ∧ (v ∩ x) ⊆ (F ‘suc u))))
42	38, 41	syl5bir 184	. . . . 5 ⊢ (u ∈ ω → ((F ‘u) ⊆ (F ‘suc u) → (v ∈ (F ‘suc u) → v ∈ (F ‘suc suc u))))
43	42	imp 277	. . . 4 ⊢ ((u ∈ ω ∧ (F ‘u) ⊆ (F ‘suc u)) → (v ∈ (F ‘suc u) → v ∈ (F ‘suc suc u)))
44	43	ssrdv 1509	. . 3 ⊢ ((u ∈ ω ∧ (F ‘u) ⊆ (F ‘suc u)) → (F ‘suc u) ⊆ (F ‘suc suc u))
45	44	exp 291	. 2 ⊢ (u ∈ ω → ((F ‘u) ⊆ (F ‘suc u) → (F ‘suc u) ⊆ (F ‘suc suc u)))
46	4, 8, 12, 16, 22, 45	finds 2397	1 ⊢ (A ∈ ω → (F ‘A) ⊆ (F ‘suc A))

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  {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:  inf3lem4 3467

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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