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Theorem isfin3 9069
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 9061 . . 3 FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
21eleq2i 2690 . 2 (𝐴 ∈ FinIII𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3 elex 3201 . . . 4 (𝒫 𝐴 ∈ FinIV → 𝒫 𝐴 ∈ V)
4 pwexb 6929 . . . 4 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
53, 4sylibr 224 . . 3 (𝒫 𝐴 ∈ FinIV𝐴 ∈ V)
6 pweq 4138 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
76eleq1d 2683 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
85, 7elab3 3345 . 2 (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
92, 8bitri 264 1 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3189  𝒫 cpw 4135  FinIVcfin4 9053  FinIIIcfin3 9054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408  df-fin3 9061
This theorem is referenced by:  fin23lem41  9125  isfin32i  9138  fin34  9163
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