Theorem pwexr 7765

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5338. (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 5433	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7740	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2868	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455  𝒫 cpw 4563  ∪ cuni 4873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911  df-ss 3923  df-pw 4565  df-sn 4591  df-pr 4593  df-uni 4874

This theorem is referenced by:  pwexb  7766  pwuninelOLD  8273  pwwf  9780  r1pw  9818  isfin3  10281  dis2ndc  23598  numufl  24053  bj-discrmoore  37734