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Theorem ssconb 3106
Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
ssconb  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( A  C_  ( C  \  B )  <->  B  C_  ( C  \  A ) ) )

Proof of Theorem ssconb
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 2994 . . . . . . 7  |-  ( A 
C_  C  ->  (
x  e.  A  ->  x  e.  C )
)
2 ssel 2994 . . . . . . 7  |-  ( B 
C_  C  ->  (
x  e.  B  ->  x  e.  C )
)
3 pm5.1 566 . . . . . . 7  |-  ( ( ( x  e.  A  ->  x  e.  C )  /\  ( x  e.  B  ->  x  e.  C ) )  -> 
( ( x  e.  A  ->  x  e.  C )  <->  ( x  e.  B  ->  x  e.  C ) ) )
41, 2, 3syl2an 283 . . . . . 6  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( x  e.  A  ->  x  e.  C )  <->  ( x  e.  B  ->  x  e.  C ) ) )
5 con2b 626 . . . . . . 7  |-  ( ( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  B  ->  -.  x  e.  A ) )
65a1i 9 . . . . . 6  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( x  e.  A  ->  -.  x  e.  B )  <->  ( x  e.  B  ->  -.  x  e.  A ) ) )
74, 6anbi12d 457 . . . . 5  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( ( x  e.  A  ->  x  e.  C )  /\  (
x  e.  A  ->  -.  x  e.  B
) )  <->  ( (
x  e.  B  ->  x  e.  C )  /\  ( x  e.  B  ->  -.  x  e.  A
) ) ) )
8 jcab 568 . . . . 5  |-  ( ( x  e.  A  -> 
( x  e.  C  /\  -.  x  e.  B
) )  <->  ( (
x  e.  A  ->  x  e.  C )  /\  ( x  e.  A  ->  -.  x  e.  B
) ) )
9 jcab 568 . . . . 5  |-  ( ( x  e.  B  -> 
( x  e.  C  /\  -.  x  e.  A
) )  <->  ( (
x  e.  B  ->  x  e.  C )  /\  ( x  e.  B  ->  -.  x  e.  A
) ) )
107, 8, 93bitr4g 221 . . . 4  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( x  e.  A  ->  ( x  e.  C  /\  -.  x  e.  B ) )  <->  ( x  e.  B  ->  ( x  e.  C  /\  -.  x  e.  A )
) ) )
11 eldif 2983 . . . . 5  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
1211imbi2i 224 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( C  \  B ) )  <->  ( x  e.  A  ->  ( x  e.  C  /\  -.  x  e.  B )
) )
13 eldif 2983 . . . . 5  |-  ( x  e.  ( C  \  A )  <->  ( x  e.  C  /\  -.  x  e.  A ) )
1413imbi2i 224 . . . 4  |-  ( ( x  e.  B  ->  x  e.  ( C  \  A ) )  <->  ( x  e.  B  ->  ( x  e.  C  /\  -.  x  e.  A )
) )
1510, 12, 143bitr4g 221 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( ( x  e.  A  ->  x  e.  ( C  \  B ) )  <->  ( x  e.  B  ->  x  e.  ( C  \  A ) ) ) )
1615albidv 1746 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( A. x ( x  e.  A  ->  x  e.  ( C  \  B ) )  <->  A. x
( x  e.  B  ->  x  e.  ( C 
\  A ) ) ) )
17 dfss2 2989 . 2  |-  ( A 
C_  ( C  \  B )  <->  A. x
( x  e.  A  ->  x  e.  ( C 
\  B ) ) )
18 dfss2 2989 . 2  |-  ( B 
C_  ( C  \  A )  <->  A. x
( x  e.  B  ->  x  e.  ( C 
\  A ) ) )
1916, 17, 183bitr4g 221 1  |-  ( ( A  C_  C  /\  B  C_  C )  -> 
( A  C_  ( C  \  B )  <->  B  C_  ( C  \  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    e. wcel 1434    \ cdif 2971    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
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