Theorem isfin3 10210

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 10202	. . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
2	1	eleq2i 2829	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3		pwexr 7712	. . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V)
4		pweq 4569	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
5	4	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
6	3, 5	elab3 3642	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
7	2, 6	bitri 275	1 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3441  𝒫 cpw 4555  Fin^IVcfin4 10194  Fin^IIIcfin3 10195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-un 3907  df-ss 3919  df-pw 4557  df-sn 4582  df-pr 4584  df-uni 4865  df-fin3 10202

This theorem is referenced by:  fin23lem41  10266  isfin32i  10279  fin34  10304