Theorem isfin3 10365

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 10357	. . 3 ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV}
2	1	eleq2i 2836	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝐴 ∈ {𝑥 ∣ 𝒫 𝑥 ∈ FinIV})
3		pwexr 7800	. . 3 ⊢ (𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V)
4		pweq 4636	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
5	4	eleq1d 2829	. . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ FinIV ↔ 𝒫 𝐴 ∈ FinIV))
6	3, 5	elab3 3702	. 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 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488  𝒫 cpw 4622  Fin^IVcfin4 10349  Fin^IIIcfin3 10350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932  df-fin3 10357

This theorem is referenced by:  fin23lem41  10421  isfin32i  10434  fin34  10459