Theorem pwid 4521

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 4519	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ 𝒫 𝐴

Syntax hints:   ∈ wcel 2111  Vcvv 3441  𝒫 cpw 4497

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499

This theorem is referenced by:  pwnex  7461  r1ordg  9191  rankr1id  9275  cfss  9676  0ram  16346  evl1fval1lem  20954  bastg  21571  fincmp  21998  restlly  22088  ptbasfi  22186  zfbas  22501  ustfilxp  22818  minveclem3b  24032  wilthlem3  25655  coinflipprob  31847  mapdunirnN  38946  pwtrrVD  41531  vsetrec  45232