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Theorem elpwi 3514
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 3513 . 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 1480    C_ wss 3066   ~Pcpw 3505
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507
This theorem is referenced by:  elpwid  3516  elelpwi  3517  elpw2g  4076  eldifpw  4393  iunpw  4396  f1opw2  5969  fi0  6856  cnntr  12383
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