Theorem elpwi 3573

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 3572	. 2 |- ( A e. ~P B -> ( A e. ~P B <-> A C_ B ) )
2	1	ibi 175	1 |- ( A e. ~P B -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 2141   C_ wss 3121   ~Pcpw 3564

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 3566

This theorem is referenced by:  elpwid  3575  elelpwi  3576  elpw2g  4140  eldifpw  4460  iunpw  4463  f1opw2  6052  pw1dc1  6887  fi0  6948  pw1on  7190  cnntr  12940