Theorem pwid 3605

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 3604	. 2 |- ( A e. _V -> A e. ~P A )
3	1, 2	ax-mp 5	1 |- A e. ~P A

Syntax hints:   e. wcel 2160   _Vcvv 2752   ~Pcpw 3590

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592

This theorem is referenced by:  pwnex  4464  pw1fin  6933  bastg  13998  pw1nct  15190