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Theorem disj3 3323
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 pm4.71 381 . . . 4  |-  ( ( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  <->  ( x  e.  A  /\  -.  x  e.  B ) ) )
2 eldif 2997 . . . . 5  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
32bibi2i 225 . . . 4  |-  ( ( x  e.  A  <->  x  e.  ( A  \  B ) )  <->  ( x  e.  A  <->  ( x  e.  A  /\  -.  x  e.  B ) ) )
41, 3bitr4i 185 . . 3  |-  ( ( x  e.  A  ->  -.  x  e.  B
)  <->  ( x  e.  A  <->  x  e.  ( A  \  B ) ) )
54albii 1402 . 2  |-  ( A. x ( x  e.  A  ->  -.  x  e.  B )  <->  A. x
( x  e.  A  <->  x  e.  ( A  \  B ) ) )
6 disj1 3321 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
7 dfcleq 2079 . 2  |-  ( A  =  ( A  \  B )  <->  A. x
( x  e.  A  <->  x  e.  ( A  \  B ) ) )
85, 6, 73bitr4i 210 1  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1285    = wceq 1287    e. wcel 1436    \ cdif 2985    i^i cin 2987   (/)c0 3275
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-dif 2990  df-in 2994  df-nul 3276
This theorem is referenced by:  disjel  3325  uneqdifeqim  3355  difprsn1  3561  diftpsn3  3563  orddif  4338  phpm  6535
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