Theorem pwid 4619

Description: A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
pwid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
pwid	⊢ 𝐴 ∈ 𝒫 𝐴

Proof of Theorem pwid

Step	Hyp	Ref	Expression
1		pwid.1	. 2 ⊢ 𝐴 ∈ V
2		pwidg 4617	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ 𝒫 𝐴

Syntax hints:   ∈ wcel 2099  Vcvv 3462  𝒫 cpw 4597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ss 3963  df-pw 4599

This theorem is referenced by:  pwnex  7759  r1ordg  9814  rankr1id  9898  cfss  10299  0ram  17017  evl1fval1lem  22318  bastg  22957  fincmp  23385  restlly  23475  ptbasfi  23573  zfbas  23888  ustfilxp  24205  minveclem3b  25444  wilthlem3  27095  coinflipprob  34326  mapdunirnN  41362  pwtrrVD  44538  vsetrec  48485