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Theorem elpwi 3575
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
StepHypRef Expression
1 elpwg 3574 . 2  |-  ( A  e.  ~P B  -> 
( A  e.  ~P B 
<->  A  C_  B )
)
21ibi 175 1  |-  ( A  e.  ~P B  ->  A  C_  B )
Colors of variables: wff set class
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:  elpwid  3577  elelpwi  3578  elpw2g  4142  eldifpw  4462  iunpw  4465  f1opw2  6055  pw1dc1  6891  fi0  6952  pw1on  7203  cnntr  13019
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