Theorem pwid 3558

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 e. _V

Assertion

Ref	Expression
pwid	|- A e. ~P A

Proof of Theorem pwid

Step	Hyp	Ref	Expression
1		pwid.1	. 2 |- A e. _V
2		pwidg 3557	. 2 |- ( A e. _V -> A e. ~P A )
3	1, 2	ax-mp 5	1 |- A e. ~P A

Syntax hints:   e. wcel 2128   _Vcvv 2712   ~Pcpw 3543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545

This theorem is referenced by:  pwnex  4407  pw1fin  6848  bastg  12421  pw1nct  13535