Theorem pwid 3668

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 3667	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ 𝒫 𝐴

Syntax hints:   ∈ wcel 2201  Vcvv 2801  𝒫 cpw 3653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-in 3205  df-ss 3212  df-pw 3655

This theorem is referenced by:  pwnex  4548  pw1fin  7107  bastg  14814  pw1nct  16664  pw1dceq  16665  exmidcon  16667