Theorem pwexr 7748

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5362. (Contributed by NM, 11-Nov-2003.)

Assertion

Ref	Expression
pwexr	⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Proof of Theorem pwexr

Step	Hyp	Ref	Expression
1		unipw 5449	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7726	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2838	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3474  𝒫 cpw 4601  ∪ cuni 4907

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3952  df-in 3954  df-ss 3964  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908

This theorem is referenced by:  pwexb  7749  pwuninel  8256  pwfiOLD  9343  pwwf  9798  r1pw  9836  isfin3  10287  dis2ndc  22955  numufl  23410  bj-discrmoore  35980