Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  conss34 Unicode version

Theorem conss34 27645
Description: Contrpositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
conss34  |-  ( A 
C_  B  <->  ( _V  \  B )  C_  ( _V  \  A ) )

Proof of Theorem conss34
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 con34b 283 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( -.  x  e.  B  ->  -.  x  e.  A
) )
2 compel 27640 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
3 compel 27640 . . . . 5  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
42, 3imbi12i 316 . . . 4  |-  ( ( x  e.  ( _V 
\  B )  ->  x  e.  ( _V  \  A ) )  <->  ( -.  x  e.  B  ->  -.  x  e.  A ) )
51, 4bitr4i 243 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  ( _V 
\  B )  ->  x  e.  ( _V  \  A ) ) )
65albii 1553 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  ( _V  \  B )  ->  x  e.  ( _V  \  A ) ) )
7 dfss2 3169 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
8 dfss2 3169 . 2  |-  ( ( _V  \  B ) 
C_  ( _V  \  A )  <->  A. x
( x  e.  ( _V  \  B )  ->  x  e.  ( _V  \  A ) ) )
96, 7, 83bitr4i 268 1  |-  ( A 
C_  B  <->  ( _V  \  B )  C_  ( _V  \  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527    e. wcel 1684   _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
  Copyright terms: Public domain W3C validator