Theorem pwexr 7693

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5301. (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 5389	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7668	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2834	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2110  Vcvv 3434  𝒫 cpw 4548  ∪ cuni 4857

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 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3436  df-un 3905  df-ss 3917  df-pw 4550  df-sn 4575  df-pr 4577  df-uni 4858

This theorem is referenced by:  pwexb  7694  pwuninel  8200  pwwf  9692  r1pw  9730  isfin3  10179  dis2ndc  23368  numufl  23823  bj-discrmoore  37124