Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  conss2 Unicode version

Theorem conss2 27315
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 3312 . 2  |-  A  C_  _V
2 ssv 3312 . 2  |-  B  C_  _V
3 ssconb 3424 . 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 2900    \ cdif 3261    C_ wss 3264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-dif 3267  df-in 3271  df-ss 3278
  Copyright terms: Public domain W3C validator