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

Theorem xpsndisj 5430
Description: Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
Assertion
Ref Expression
xpsndisj  |-  ( B  =/=  D  ->  (
( A  X.  { B } )  i^i  ( C  X.  { D }
) )  =  (/) )

Proof of Theorem xpsndisj
StepHypRef Expression
1 disjsn2 4089 . 2  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2 xpdisj2 5429 . 2  |-  ( ( { B }  i^i  { D } )  =  (/)  ->  ( ( A  X.  { B }
)  i^i  ( C  X.  { D } ) )  =  (/) )
31, 2syl 16 1  |-  ( B  =/=  D  ->  (
( A  X.  { B } )  i^i  ( C  X.  { D }
) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    =/= wne 2662    i^i cin 3475   (/)c0 3785   {csn 4027    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007
This theorem is referenced by:  xp01disj  7147  unxpdom2  7729  sucxpdom  7730
  Copyright terms: Public domain W3C validator