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Theorem disj 3855
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 3468 . . . 4  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
21eqeq1i 2461 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  { x  |  ( x  e.  A  /\  x  e.  B ) }  =  (/) )
3 abeq1 2579 . . 3  |-  ( { x  |  ( x  e.  A  /\  x  e.  B ) }  =  (/)  <->  A. x ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  (/) ) )
4 imnan 420 . . . . 5  |-  ( ( x  e.  A  ->  -.  x  e.  B
)  <->  -.  ( x  e.  A  /\  x  e.  B ) )
5 noel 3787 . . . . . 6  |-  -.  x  e.  (/)
65nbn 345 . . . . 5  |-  ( -.  ( x  e.  A  /\  x  e.  B
)  <->  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  (/) ) )
74, 6bitr2i 250 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  B
)  <->  x  e.  (/) )  <->  ( x  e.  A  ->  -.  x  e.  B ) )
87albii 1645 . . 3  |-  ( A. x ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  (/) )  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
92, 3, 83bitri 271 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
10 df-ral 2809 . 2  |-  ( A. x  e.  A  -.  x  e.  B  <->  A. x
( x  e.  A  ->  -.  x  e.  B
) )
119, 10bitr4i 252 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804    i^i cin 3460   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-v 3108  df-dif 3464  df-in 3468  df-nul 3784
This theorem is referenced by:  disjr  3856  disj1  3857  disjne  3860  otiunsndisj  4742  onxpdisj  5071  f0rn0  5752  onint  6603  zfreg  8013  kmlem4  8524  fin23lem30  8713  fin23lem31  8714  isf32lem3  8726  fpwwe2  9010  renfdisj  9636  injresinjlem  11906  metdsge  21522  spthispth  24780  2spotdisj  25266  2spotiundisj  25267  2spotmdisj  25273  subfacp1lem1  28890  dfpo2  29428  dvmptfprodlem  31983  stoweidlem26  32050  stoweidlem59  32083  otiunsndisjX  32694
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