MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disj1 Unicode version

Theorem disj1 3630
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 3628 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 df-ral 2671 . 2  |-  ( A. x  e.  A  -.  x  e.  B  <->  A. x
( x  e.  A  ->  -.  x  e.  B
) )
31, 2bitri 241 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 177   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2666    i^i cin 3279   (/)c0 3588
This theorem is referenced by:  reldisj  3631  disj3  3632  undif4  3644  disjsn  3828  funun  5454  zfregs2  7625  dfac5lem4  7963  isf32lem9  8197  fzodisj  11122  zfregs2VD  28662  bnj1280  29095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589
  Copyright terms: Public domain W3C validator