Theorem pwid 4580

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 4577	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ 𝒫 𝐴

This proof depends on syntax axioms:   ∈ wcel 2144  Vcvv 3450  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-pw 4558

This theorem is used by:  pwnex  7773  r1ordg  9788  r1wf  9844  rankr1id  9881  cfss  10346  0ram  17203  evl1fval1lem  22653  bastg  23289  fincmp  23716  restlly  23807  ptbasfi  23905  zfbas  24220  ustfilxp  24537  minveclem3b  25754  wilthlem3  27402  coinflipprob  35158  mapdunirnN  42800  pwtrrVD  45900  vsetrec  50964