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Theorem reldisj 3815
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 3433 . . . 4  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 pm5.44 914 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  C )  ->  ( ( x  e.  A  ->  -.  x  e.  B )  <->  ( x  e.  A  ->  ( x  e.  C  /\  -.  x  e.  B )
) ) )
3 eldif 3426 . . . . . . 7  |-  ( x  e.  ( C  \  B )  <->  ( x  e.  C  /\  -.  x  e.  B ) )
43imbi2i 312 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  ( C  \  B ) )  <->  ( x  e.  A  ->  ( x  e.  C  /\  -.  x  e.  B )
) )
52, 4syl6bbr 265 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  C )  ->  ( ( x  e.  A  ->  -.  x  e.  B )  <->  ( x  e.  A  ->  x  e.  ( C  \  B
) ) ) )
65sps 1891 . . . 4  |-  ( A. x ( x  e.  A  ->  x  e.  C )  ->  (
( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  ->  x  e.  ( C  \  B ) ) ) )
71, 6sylbi 197 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  ->  x  e.  ( C  \  B ) ) ) )
87albidv 1736 . 2  |-  ( A 
C_  C  ->  ( A. x ( x  e.  A  ->  -.  x  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  ( C 
\  B ) ) ) )
9 disj1 3814 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
10 dfss2 3433 . 2  |-  ( A 
C_  ( C  \  B )  <->  A. x
( x  e.  A  ->  x  e.  ( C 
\  B ) ) )
118, 9, 103bitr4g 290 1  |-  ( A 
C_  C  ->  (
( A  i^i  B
)  =  (/)  <->  A  C_  ( C  \  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369   A.wal 1405    = wceq 1407    e. wcel 1844    \ cdif 3413    i^i cin 3415    C_ wss 3416   (/)c0 3740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-v 3063  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3741
This theorem is referenced by:  disj2  3819  oacomf1olem  7252  domdifsn  7640  elfiun  7926  cantnfp1lem3  8133  cantnfp1lem3OLD  8159  ssxr  9687  structcnvcnv  14854  fidomndrng  18278  elcls  19869  ist1-2  20143  nrmsep2  20152  nrmsep  20153  isnrm3  20155  isreg2  20173  hauscmplem  20201  connsub  20216  iunconlem  20222  llycmpkgen2  20345  hausdiag  20440  trfil3  20683  isufil2  20703  filufint  20715  blcld  21302  i1fima2  22380  i1fd  22382  usgrares1  24839  nbgrassvwo  24866  nbgrassvwo2  24867  itg2addnclem2  31453
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