Theorem pwexr 7773

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5369. (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 5456	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7751	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2834	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3473  𝒫 cpw 4606  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-un 3954  df-in 3956  df-ss 3966  df-pw 4608  df-sn 4633  df-pr 4635  df-uni 4913

This theorem is referenced by:  pwexb  7774  pwuninel  8287  pwfiOLD  9379  pwwf  9838  r1pw  9876  isfin3  10327  dis2ndc  23384  numufl  23839  bj-discrmoore  36623