Theorem pwexr 7800

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5383. (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 5470	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7775	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2849	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3488  𝒫 cpw 4622  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932

This theorem is referenced by:  pwexb  7801  pwuninel  8316  pwfiOLD  9417  pwwf  9876  r1pw  9914  isfin3  10365  dis2ndc  23489  numufl  23944  bj-discrmoore  37077