MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin3 Structured version   Visualization version   GIF version

Theorem isfin3 10334
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 10326 . . 3 FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
21eleq2i 2831 . 2 (𝐴 ∈ FinIII𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3 pwexr 7784 . . 3 (𝒫 𝐴 ∈ FinIV𝐴 ∈ V)
4 pweq 4619 . . . 4 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
54eleq1d 2824 . . 3 (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
63, 5elab3 3689 . 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 206   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  𝒫 cpw 4605  FinIVcfin4 10318  FinIIIcfin3 10319
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-fin3 10326
This theorem is referenced by:  fin23lem41  10390  isfin32i  10403  fin34  10428
  Copyright terms: Public domain W3C validator