Theorem isfin3 10320

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.

Step	Hyp	Ref	Expression
1		df-fin3 10312	. . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
2	1	eleq2i 2821	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3		pwexr 7767	. . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V)
4		pweq 4617	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
5	4	eleq1d 2814	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
6	3, 5	elab3 3675	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
7	2, 6	bitri 275	1 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)

Syntax hints:   ↔ wb 205   = wceq 1534   ∈ wcel 2099  {cab 2705  Vcvv 3471  𝒫 cpw 4603  Fin^IVcfin4 10304  Fin^IIIcfin3 10305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-un 3952  df-in 3954  df-ss 3964  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4909  df-fin3 10312

This theorem is referenced by:  fin23lem41  10376  isfin32i  10389  fin34  10414