Theorem elpwi 3610

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 3609	. 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 2164   C_ wss 3153   ~Pcpw 3601

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by:  elpwid  3612  elelpwi  3613  elpw2g  4185  eldifpw  4508  iunpw  4511  f1opw2  6124  pw1dc1  6970  fi0  7034  pw1on  7286  lspf  13885  cnntr  14393