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Theorem isfin3 9712
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 9704 . . 3 FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
21eleq2i 2904 . 2 (𝐴 ∈ FinIII𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3 pwexr 7481 . . 3 (𝒫 𝐴 ∈ FinIV𝐴 ∈ V)
4 pweq 4542 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
54eleq1d 2897 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
63, 5elab3 3674 . 2 (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
72, 6bitri 277 1 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  wb 208   = wceq 1533  wcel 2110  {cab 2799  𝒫 cpw 4539  FinIVcfin4 9696  FinIIIcfin3 9697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-pw 4541  df-sn 4562  df-pr 4564  df-uni 4833  df-fin3 9704
This theorem is referenced by:  fin23lem41  9768  isfin32i  9781  fin34  9806
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