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Theorem reldisj 3727
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 3350 . . . 4  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 pm5.44 902 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  C )  ->  ( ( x  e.  A  ->  -.  x  e.  B )  <->  ( x  e.  A  ->  ( x  e.  C  /\  -.  x  e.  B )
) ) )
3 eldif 3343 . . . . . . 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 263 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  C )  ->  ( ( x  e.  A  ->  -.  x  e.  B )  <->  ( x  e.  A  ->  x  e.  ( C  \  B
) ) ) )
65sps 1800 . . . 4  |-  ( A. x ( x  e.  A  ->  x  e.  C )  ->  (
( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  ->  x  e.  ( C  \  B ) ) ) )
71, 6sylbi 195 . . 3  |-  ( A 
C_  C  ->  (
( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  ->  x  e.  ( C  \  B ) ) ) )
87albidv 1679 . 2  |-  ( A 
C_  C  ->  ( A. x ( x  e.  A  ->  -.  x  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  ( C 
\  B ) ) ) )
9 disj1 3726 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
10 dfss2 3350 . 2  |-  ( A 
C_  ( C  \  B )  <->  A. x
( x  e.  A  ->  x  e.  ( C 
\  B ) ) )
118, 9, 103bitr4g 288 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 184    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756    \ cdif 3330    i^i cin 3332    C_ wss 3333   (/)c0 3642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-v 2979  df-dif 3336  df-in 3340  df-ss 3347  df-nul 3643
This theorem is referenced by:  disj2  3731  oacomf1olem  7008  domdifsn  7399  elfiun  7685  cantnfp1lem3  7893  cantnfp1lem3OLD  7919  ssxr  9449  structcnvcnv  14190  fidomndrng  17384  elcls  18682  ist1-2  18956  nrmsep2  18965  nrmsep  18966  isnrm3  18968  isreg2  18986  hauscmplem  19014  connsub  19030  iunconlem  19036  llycmpkgen2  19128  hausdiag  19223  trfil3  19466  isufil2  19486  filufint  19498  blcld  20085  i1fima2  21162  i1fd  21164  usgrares1  23328  itg2addnclem2  28449  nbgrassvwo  30281  nbgrassvwo2  30282
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