Theorem elpwi 3584

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 3583	. 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 2148   C_ wss 3129   ~Pcpw 3575

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577

This theorem is referenced by:  elpwid  3586  elelpwi  3587  elpw2g  4155  eldifpw  4476  iunpw  4479  f1opw2  6073  pw1dc1  6909  fi0  6970  pw1on  7221  cnntr  13587