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Theorem disj3 3499
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
disj3  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )

Proof of Theorem disj3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pm4.71 611 . . . 4  |-  ( ( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  <->  ( x  e.  A  /\  -.  x  e.  B ) ) )
2 eldif 3162 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
32bibi2i 304 . . . 4  |-  ( ( x  e.  A  <->  x  e.  ( A  \  B ) )  <->  ( x  e.  A  <->  ( x  e.  A  /\  -.  x  e.  B ) ) )
41, 3bitr4i 243 . . 3  |-  ( ( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  <->  x  e.  ( A  \  B ) ) )
54albii 1553 . 2  |-  ( A. x ( x  e.  A  ->  -.  x  e.  B )  <->  A. x
( x  e.  A  <->  x  e.  ( A  \  B ) ) )
6 disj1 3497 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
7 dfcleq 2277 . 2  |-  ( A  =  ( A  \  B )  <->  A. x
( x  e.  A  <->  x  e.  ( A  \  B ) ) )
85, 6, 73bitr4i 268 1  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151   (/)c0 3455
This theorem is referenced by:  disjel  3501  disj4  3503  uneqdifeq  3542  ssunsn2  3773  orddif  4486  php  7045  hartogslem1  7257  infeq5i  7337  cantnfp1lem3  7382  cda1dif  7802  infcda1  7819  ssxr  8892  dprd2da  15277  dmdprdsplit2lem  15280  ablfac1eulem  15307  lbsextlem4  15914  opsrtoslem2  16226  alexsublem  17738  volun  18902  lhop1lem  19360  ex-dif  20810  ballotlemfp1  23050  difeq  23128  difprsn2  23191  disjdsct  23369  probun  23622  kelac2  27163  pwfi2f1o  27260  difprsneq  28068  difprsng  28069  diftpsneq  28070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-nul 3456
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