Theorem pwexr 7704

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5305. (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 5393	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7679	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2838	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3437  𝒫 cpw 4549  ∪ cuni 4858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903  df-ss 3915  df-pw 4551  df-sn 4576  df-pr 4578  df-uni 4859

This theorem is referenced by:  pwexb  7705  pwuninel  8211  pwwf  9707  r1pw  9745  isfin3  10194  dis2ndc  23376  numufl  23831  bj-discrmoore  37176