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Theorem isfin3 10290
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 10282 . . 3 FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
21eleq2i 2819 . 2 (𝐴 ∈ FinIII𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3 pwexr 7748 . . 3 (𝒫 𝐴 ∈ FinIV𝐴 ∈ V)
4 pweq 4611 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
54eleq1d 2812 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
63, 5elab3 3671 . 2 (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
72, 6bitri 275 1 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468  𝒫 cpw 4597  FinIVcfin4 10274  FinIIIcfin3 10275
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-ext 2697  ax-sep 5292  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4903  df-fin3 10282
This theorem is referenced by:  fin23lem41  10346  isfin32i  10359  fin34  10384
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