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Theorem conss34 27748
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 27743 . . . . 5  |-  ( x  e.  ( _V  \  B )  <->  -.  x  e.  B )
3 compel 27743 . . . . 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 1556 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  ( _V  \  B )  ->  x  e.  ( _V  \  A ) ) )
7 dfss2 3182 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
8 dfss2 3182 . 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 1530    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179
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