Theorem pwexr 7811

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5334. (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 7789	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2869	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454  𝒫 cpw 4560  ∪ cuni 4872

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-pw 4562  df-sn 4589  df-pr 4591  df-uni 4873

This theorem is used by:  pwexb  7812  pwuninelOLD  8331  pwwf  9852  r1pw  9896  isfin3  10413  dis2ndc  23834  numufl  24289  bj-discrmoore  38302