Theorem pwidg 3601

Description: Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
pwidg	|- ( A e. V -> A e. ~P A )

Proof of Theorem pwidg

Step	Hyp	Ref	Expression
1		ssid 3187	. 2 |- A C_ A
2		elpwg 3595	. 2 |- ( A e. V -> ( A e. ~P A <-> A C_ A ) )
3	1, 2	mpbiri 168	1 |- ( A e. V -> A e. ~P A )

Syntax hints:   -> wi 4   e. wcel 2158   C_ wss 3141   ~Pcpw 3587

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589

This theorem is referenced by:  pwid  3602  axpweq  4183  baspartn  13821  epttop  13861  isopn3  13896