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Theorem disj 3486
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
disj  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
Distinct variable groups:    x, A    x, B

Proof of Theorem disj
StepHypRef Expression
1 df-in 3150 . . . 4  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
21eqeq1i 2197 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  (/) )
3 abeq1 2299 . . 3  |-  ( { x  |  ( x  e.  A  /\  x  e.  B ) }  =  (/)  <->  A. x ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  (/) ) )
4 imnan 691 . . . . 5  |-  ( ( x  e.  A  ->  -.  x  e.  B
)  <->  -.  ( x  e.  A  /\  x  e.  B ) )
5 noel 3441 . . . . . 6  |-  -.  x  e.  (/)
65nbn 700 . . . . 5  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  <->  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  (/) ) )
74, 6bitr2i 185 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  <->  x  e.  (/) )  <->  ( x  e.  A  ->  -.  x  e.  B ) )
87albii 1481 . . 3  |-  ( A. x ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  (/) )  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
92, 3, 83bitri 206 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
10 df-ral 2473 . 2  |-  ( A. x  e.  A  -.  x  e.  B  <->  A. x
( x  e.  A  ->  -.  x  e.  B
) )
119, 10bitr4i 187 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468    i^i cin 3143   (/)c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438
This theorem is referenced by:  disjr  3487  disj1  3488  disjne  3491  f0rn0  5426  renfdisj  8042  fvinim0ffz  10266  fxnn0nninf  10464  fprodsplitdc  11631  exmidunben  12472  dedekindeulemuub  14532  dedekindeulemlu  14536  dedekindicclemuub  14541  dedekindicclemlu  14545  ivthinclemdisj  14555  exmidsbthrlem  15208
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