Theorem pwid 4563

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 4561	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ 𝒫 𝐴

Syntax hints:   ∈ wcel 2114  Vcvv 3429  𝒫 cpw 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ss 3906  df-pw 4543

This theorem is referenced by:  pwnex  7713  r1ordg  9702  rankr1id  9786  cfss  10187  0ram  16991  evl1fval1lem  22295  bastg  22931  fincmp  23358  restlly  23448  ptbasfi  23546  zfbas  23861  ustfilxp  24178  minveclem3b  25395  wilthlem3  27033  coinflipprob  34624  r1wf  35239  mapdunirnN  42096  pwtrrVD  45251  vsetrec  50178