Theorem pwexr 7767

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5345. (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 5435	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7742	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2838	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3463  𝒫 cpw 4580  ∪ cuni 4887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-un 3936  df-ss 3948  df-pw 4582  df-sn 4607  df-pr 4609  df-uni 4888

This theorem is referenced by:  pwexb  7768  pwuninel  8282  pwwf  9829  r1pw  9867  isfin3  10318  dis2ndc  23414  numufl  23869  bj-discrmoore  37071