Theorem pwexr 7635

Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5291. (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 5369	. 2 ⊢ ∪ 𝒫 𝐴 = 𝐴
2		uniexg 7613	. 2 ⊢ (𝒫 𝐴 ∈ 𝑉 → ∪ 𝒫 𝐴 ∈ V)
3	1, 2	eqeltrrid 2839	1 ⊢ (𝒫 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2101  Vcvv 3434  𝒫 cpw 4536  ∪ cuni 4841

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 2103  ax-9 2111  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-pw 4538  df-sn 4565  df-pr 4567  df-uni 4842

This theorem is referenced by:  pwexb  7636  pwuninel  8111  pwfiOLD  9142  pwwf  9593  r1pw  9631  isfin3  10080  dis2ndc  22639  numufl  23094  bj-discrmoore  35310