Theorem pwexr 7777

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5327. (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 5418	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7755	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2866	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451  𝒫 cpw 4557  ∪ cuni 4867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-pw 4559  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is used by:  pwexb  7778  pwuninelOLD  8286  pwwf  9808  r1pw  9852  isfin3  10367  dis2ndc  23772  numufl  24227  bj-discrmoore  38012