Theorem pwexr 7764

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5330. (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 5425	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7742	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2865	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3450  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 2732  ax-sep 5251  ax-pr 5398  ax-un 7736

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 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-ss 3916  df-pw 4559  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is used by:  pwexb  7765  pwuninelOLD  8274  pwwf  9789  r1pw  9827  isfin3  10298  dis2ndc  23686  numufl  24141  bj-discrmoore  37861