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Theorem xp01disj 6048
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 6047 . . 3  |-  1o  =/=  (/)
21necomi 2305 . 2  |-  (/)  =/=  1o
3 xpsndisj 4777 . 2  |-  ( (/)  =/=  1o  ->  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o } ) )  =  (/) )
42, 3ax-mp 7 1  |-  ( ( A  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1259    =/= wne 2220    i^i cin 2944   (/)c0 3252   {csn 3403    X. cxp 4371   1oc1o 6025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-1o 6032
This theorem is referenced by:  endisj  6329
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