Theorem pwidg 3519

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 3112	. 2 |- A C_ A
2		elpwg 3513	. 2 |- ( A e. V -> ( A e. ~P A <-> A C_ A ) )
3	1, 2	mpbiri 167	1 |- ( A e. V -> A e. ~P A )

Syntax hints:   -> wi 4   e. wcel 1480   C_ wss 3066   ~Pcpw 3505

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507

This theorem is referenced by:  pwid  3520  axpweq  4090  baspartn  12206  epttop  12248  isopn3  12283