Theorem pwid 3641

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 3640	. 2 |- ( A e. _V -> A e. ~P A )
3	1, 2	ax-mp 5	1 |- A e. ~P A

Syntax hints:   e. wcel 2178   _Vcvv 2776   ~Pcpw 3626

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  pwnex  4514  pw1fin  7033  bastg  14648  pw1nct  16142