Theorem pwexr 7712

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5302. (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 5397	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7687	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2842	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3430  𝒫 cpw 4542  ∪ cuni 4851

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 5231  ax-pr 5370  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 3432  df-un 3895  df-ss 3907  df-pw 4544  df-sn 4569  df-pr 4571  df-uni 4852

This theorem is referenced by:  pwexb  7713  pwuninel  8218  pwwf  9722  r1pw  9760  isfin3  10209  dis2ndc  23435  numufl  23890  bj-discrmoore  37439