Theorem elpwi 3624

Description: Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
elpwi	|- ( A e. ~P B -> A C_ B )

Proof of Theorem elpwi

Step	Hyp	Ref	Expression
1		elpwg 3623	. 2 |- ( A e. ~P B -> ( A e. ~P B <-> A C_ B ) )
2	1	ibi 176	1 |- ( A e. ~P B -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 2175   C_ wss 3165   ~Pcpw 3615

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617

This theorem is referenced by:  elpwid  3626  elelpwi  3627  elpw2g  4199  eldifpw  4523  iunpw  4526  f1opw2  6151  pw1dc1  7010  fi0  7076  pw1on  7337  lspf  14122  cnntr  14668  2omap  15894