Theorem pwexr 7760

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5334. (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 5429	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7735	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2874	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463  𝒫 cpw 4564  ∪ cuni 4873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4874

This theorem is referenced by:  pwexb  7761  pwuninelOLD  8268  pwwf  9775  r1pw  9813  isfin3  10276  dis2ndc  23582  numufl  24037  bj-discrmoore  37636