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Theorem isfin3 9319
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 9311 . . 3 FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
21eleq2i 2841 . 2 (𝐴 ∈ FinIII𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3 pwexr 7120 . . 3 (𝒫 𝐴 ∈ FinIV𝐴 ∈ V)
4 pweq 4298 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
54eleq1d 2834 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
63, 5elab3 3507 . 2 (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
72, 6bitri 264 1 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1630  wcel 2144  {cab 2756  𝒫 cpw 4295  FinIVcfin4 9303  FinIIIcfin3 9304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-pw 4297  df-sn 4315  df-pr 4317  df-uni 4573  df-fin3 9311
This theorem is referenced by:  fin23lem41  9375  isfin32i  9388  fin34  9413
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