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Theorem disj1 3671
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 3669 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 df-ral 2711 . 2  |-  ( A. x  e.  A  -.  x  e.  B  <->  A. x
( x  e.  A  ->  -.  x  e.  B
) )
31, 2bitri 242 1  |-  ( ( A  i^i  B )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2706    i^i cin 3320   (/)c0 3629
This theorem is referenced by:  reldisj  3672  disj3  3673  undif4  3685  disjsn  3869  funun  5496  zfregs2  7670  dfac5lem4  8008  isf32lem9  8242  fzodisj  11168  zfregs2VD  28954  bnj1280  29390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-v 2959  df-dif 3324  df-in 3328  df-nul 3630
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