Theorem pwid 4576

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 4574	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ 𝒫 𝐴

Syntax hints:   ∈ wcel 2113  Vcvv 3440  𝒫 cpw 4554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ss 3918  df-pw 4556

This theorem is referenced by:  pwnex  7704  r1ordg  9690  rankr1id  9774  cfss  10175  0ram  16948  evl1fval1lem  22274  bastg  22910  fincmp  23337  restlly  23427  ptbasfi  23525  zfbas  23840  ustfilxp  24157  minveclem3b  25384  wilthlem3  27036  coinflipprob  34637  r1wf  35252  mapdunirnN  41906  pwtrrVD  45061  vsetrec  49944