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Theorem isfin3 9001
 Description: Definition of a III-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin3 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)

Proof of Theorem isfin3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-fin3 8993 . . 3 FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
21eleq2i 2680 . 2 (𝐴 ∈ FinIII𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3 elex 3185 . . . 4 (𝒫 𝐴 ∈ FinIV → 𝒫 𝐴 ∈ V)
4 pwexb 6867 . . . 4 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
53, 4sylibr 223 . . 3 (𝒫 𝐴 ∈ FinIV𝐴 ∈ V)
6 pweq 4111 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
76eleq1d 2672 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
85, 7elab3 3327 . 2 (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
92, 8bitri 263 1 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  𝒫 cpw 4108  FinIVcfin4 8985  FinIIIcfin3 8986 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-fin3 8993 This theorem is referenced by:  fin23lem41  9057  isfin32i  9070  fin34  9095
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