Theorem pwidg 3580

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 3167	. 2 |- A C_ A
2		elpwg 3574	. 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 2141   C_ wss 3121   ~Pcpw 3566

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568

This theorem is referenced by:  pwid  3581  axpweq  4157  baspartn  12842  epttop  12884  isopn3  12919