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Theorem xp01disj 7033
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.)
Assertion
Ref Expression
xp01disj  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)

Proof of Theorem xp01disj
StepHypRef Expression
1 1n0 7032 . . 3  |-  1o  =/=  (/)
21necomi 2716 . 2  |-  (/)  =/=  1o
3 xpsndisj 5356 . 2  |-  ( (/)  =/=  1o  ->  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o } ) )  =  (/) )
42, 3ax-mp 5 1  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    =/= wne 2642    i^i cin 3422   (/)c0 3732   {csn 3972    X. cxp 4933   1oc1o 7010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-br 4388  df-opab 4446  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-1o 7017
This theorem is referenced by:  endisj  7495  uncdadom  8438  cdaun  8439  cdaen  8440  cda1dif  8443  pm110.643  8444  cdacomen  8448  cdaassen  8449  xpcdaen  8450  mapcdaen  8451  cdadom1  8453  infcda1  8460
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