Theorem pwexr 7780

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5326. (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 5417	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7758	. 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 2144  Vcvv 3450  𝒫 cpw 4556  ∪ cuni 4866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-un 4151  df-pw 4558  df-sn 4585  df-pr 4587  df-uni 4867

This theorem is used by:  pwexb  7781  pwuninelOLD  8297  pwwf  9819  r1pw  9863  isfin3  10378  dis2ndc  23785  numufl  24240  bj-discrmoore  38125