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Theorem reldisj 3311
Description: Two ways of saying that two classes are disjoint, using the complement of  B relative to a universe  C. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
reldisj  |-  ( A 
C_  C  ->  (
( A  i^i  B
)  =  (/)  <->  A  C_  ( C  \  B ) ) )

Proof of Theorem reldisj
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 2997 . . . 4  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 pm5.44 868 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  C )  ->  ( ( x  e.  A  ->  -.  x  e.  B )  <->  ( x  e.  A  ->  ( x  e.  C  /\  -.  x  e.  B )
) ) )
3 eldif 2991 . . . . . . 7  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
43imbi2i 224 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  ( C  \  B ) )  <->  ( x  e.  A  ->  ( x  e.  C  /\  -.  x  e.  B )
) )
52, 4syl6bbr 196 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  C )  ->  ( ( x  e.  A  ->  -.  x  e.  B )  <->  ( x  e.  A  ->  x  e.  ( C  \  B
) ) ) )
65sps 1471 . . . 4  |-  ( A. x ( x  e.  A  ->  x  e.  C )  ->  (
( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  ->  x  e.  ( C  \  B ) ) ) )
71, 6sylbi 119 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  ->  x  e.  ( C  \  B ) ) ) )
87albidv 1747 . 2  |-  ( A 
C_  C  ->  ( A. x ( x  e.  A  ->  -.  x  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  ( C 
\  B ) ) ) )
9 disj1 3310 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
10 dfss2 2997 . 2  |-  ( A 
C_  ( C  \  B )  <->  A. x
( x  e.  A  ->  x  e.  ( C 
\  B ) ) )
118, 9, 103bitr4g 221 1  |-  ( A 
C_  C  ->  (
( A  i^i  B
)  =  (/)  <->  A  C_  ( C  \  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285    e. wcel 1434    \ cdif 2979    i^i cin 2981    C_ wss 2982   (/)c0 3267
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-nul 3268
This theorem is referenced by:  disj2  3315  ssdifsn  3536
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