Theorem pwidg 3488

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 3081	. 2 |- A C_ A
2		elpwg 3482	. 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 1461   C_ wss 3035   ~Pcpw 3474

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476

This theorem is referenced by:  pwid  3489  axpweq  4053  baspartn  12054  epttop  12096  isopn3  12131