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Theorem conss2 27646
Description: Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
conss2  |-  ( A 
C_  ( _V  \  B )  <->  B  C_  ( _V  \  A ) )

Proof of Theorem conss2
StepHypRef Expression
1 ssv 3198 . 2  |-  A  C_  _V
2 ssv 3198 . 2  |-  B  C_  _V
3 ssconb 3309 . 2  |-  ( ( A  C_  _V  /\  B  C_ 
_V )  ->  ( A  C_  ( _V  \  B )  <->  B  C_  ( _V  \  A ) ) )
41, 2, 3mp2an 653 1  |-  ( A 
C_  ( _V  \  B )  <->  B  C_  ( _V  \  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   _Vcvv 2788    \ cdif 3149    C_ wss 3152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166
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