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Theorem conss2 27614
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 3361 . 2  |-  A  C_  _V
2 ssv 3361 . 2  |-  B  C_  _V
3 ssconb 3473 . 2  |-  ( ( A  C_  _V  /\  B  C_ 
_V )  ->  ( A  C_  ( _V  \  B )  <->  B  C_  ( _V  \  A ) ) )
41, 2, 3mp2an 654 1  |-  ( A 
C_  ( _V  \  B )  <->  B  C_  ( _V  \  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   _Vcvv 2949    \ cdif 3310    C_ wss 3313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2951  df-dif 3316  df-in 3320  df-ss 3327
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