Theorem inf3lem4 3467

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
inf3lem4	⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → (F ‘A) ⊂ (F ‘suc A)))

Distinct variable group(s):   x,y,z,w

Proof of Theorem inf3lem4

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . 5 ⊢ G = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
2		inf3lem.2	. . . . 5 ⊢ F = (rec(G, ∅) ↾ ω)
3		inf3lem.3	. . . . 5 ⊢ A ∈ V
4		inf3lem.4	. . . . 5 ⊢ B ∈ V
5	1, 2, 3, 4	inf3lem1 3464	. . . 4 ⊢ (A ∈ ω → (F ‘A) ⊆ (F ‘suc A))
6	5	a1i 7	. . 3 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → (F ‘A) ⊆ (F ‘suc A)))
7	1, 2, 3, 4	inf3lem3 3466	. . 3 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → ¬ (F ‘A) = (F ‘suc A)))
8	6, 7	jcad 455	. 2 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → ((F ‘A) ⊆ (F ‘suc A) ∧ ¬ (F ‘A) = (F ‘suc A))))
9		dfpss2 1557	. 2 ⊢ ((F ‘A) ⊂ (F ‘suc A) ↔ ((F ‘A) ⊆ (F ‘suc A) ∧ ¬ (F ‘A) = (F ‘suc A)))
10	8, 9	syl6ibr 186	1 ⊢ ((¬ x = ∅ ∧ x ⊆ ∪x) → (A ∈ ω → (F ‘A) ⊂ (F ‘suc A)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487   ⊂ wpss 1488  ∅c0 1707  ∪cuni 1919  {copab 2055  suc csuc 2201  ωcom 2372   ↾ cres 2412   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  inf3lem5 3468

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  ax-reg 1078

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-ne 1192  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-pss 1494  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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