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

Theorem disj1 3709
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 3707 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 df-ral 2710 . 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 1360    = wceq 1362    e. wcel 1755   A.wral 2705    i^i cin 3315   (/)c0 3625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-v 2964  df-dif 3319  df-in 3323  df-nul 3626
This theorem is referenced by:  reldisj  3710  disj3  3711  undif4  3723  disjsn  3924  funun  5448  zfregs2  7941  dfac5lem4  8284  isf32lem9  8518  fzodisj  11566  zfregs2VD  31276  bnj1280  31710
  Copyright terms: Public domain W3C validator