Theorem isfin3 9001

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 8993	. . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
2	1	eleq2i 2680	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3		elex 3185	. . . 4 ⊢ (𝒫 𝐴 ∈ FinIV → 𝒫 𝐴 ∈ V)
4		pwexb 6867	. . . 4 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
5	3, 4	sylibr 223	. . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V)
6		pweq 4111	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
7	6	eleq1d 2672	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
8	5, 7	elab3 3327	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} ↔ 𝒫 𝐴 ∈ FinIV)
9	2, 8	bitri 263	1 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  𝒫 cpw 4108  Fin^IVcfin4 8985  Fin^IIIcfin3 8986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-fin3 8993

This theorem is referenced by:  fin23lem41  9057  isfin32i  9070  fin34  9095