Theorem pwexr 7705

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5307. (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 7680	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2833	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3438  𝒫 cpw 4553  ∪ cuni 4861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-un 3910  df-ss 3922  df-pw 4555  df-sn 4580  df-pr 4582  df-uni 4862

This theorem is referenced by:  pwexb  7706  pwuninel  8215  pwwf  9722  r1pw  9760  isfin3  10209  dis2ndc  23363  numufl  23818  bj-discrmoore  37084