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Theorem disj1 3857
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
disj1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
Distinct variable groups:    x, A    x, B

Proof of Theorem disj1
StepHypRef Expression
1 disj 3855 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 df-ral 2809 . 2  |-  ( A. x  e.  A  -.  x  e.  B  <->  A. x
( x  e.  A  ->  -.  x  e.  B
) )
31, 2bitri 249 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1396    = wceq 1398    e. wcel 1823   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:  reldisj  3858  disj3  3859  undif4  3871  disjsn  4076  funun  5612  zfregs2  8155  dfac5lem4  8498  isf32lem9  8732  fzodisj  11836  zfregs2VD  34041  bnj1280  34477
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