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Theorem inf3lem7 9665
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9666 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7966. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
Hypotheses
Ref Expression
inf3lem.1 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
inf3lem.2 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3 𝐴 ∈ V
inf3lem.4 𝐵 ∈ V
Assertion
Ref Expression
inf3lem7 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
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 inf3lem7
StepHypRef Expression
1 inf3lem.1 . . 3 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
2 inf3lem.2 . . 3 𝐹 = (rec(𝐺, ∅) ↾ ω)
3 inf3lem.3 . . 3 𝐴 ∈ V
4 inf3lem.4 . . 3 𝐵 ∈ V
51, 2, 3, 4inf3lem6 9664 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)
6 vpwex 5381 . 2 𝒫 𝑥 ∈ V
7 f1dmex 7966 . 2 ((𝐹:ω–1-1→𝒫 𝑥 ∧ 𝒫 𝑥 ∈ V) → ω ∈ V)
85, 6, 7sylancl 584 1 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2937  {crab 3430  Vcvv 3473  cin 3948  wss 3949  c0 4326  𝒫 cpw 4606   cuni 4912  cmpt 5235  cres 5684  1-1wf1 6550  ωcom 7876  reccrdg 8436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437
This theorem is referenced by:  inf3  9666  infeq5  9668
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