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Theorem xp01disjl 6429
Description: Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
Assertion
Ref Expression
xp01disjl  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  C ) )  =  (/)

Proof of Theorem xp01disjl
StepHypRef Expression
1 1n0 6427 . . 3  |-  1o  =/=  (/)
21necomi 2432 . 2  |-  (/)  =/=  1o
3 disjsn2 3654 . 2  |-  ( (/)  =/=  1o  ->  ( { (/)
}  i^i  { 1o } )  =  (/) )
4 xpdisj1 5049 . 2  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  ->  ( ( { (/) }  X.  A )  i^i  ( { 1o }  X.  C ) )  =  (/) )
52, 3, 4mp2b 8 1  |-  ( ( { (/) }  X.  A
)  i^i  ( { 1o }  X.  C ) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1353    =/= wne 2347    i^i cin 3128   (/)c0 3422   {csn 3591    X. cxp 4621   1oc1o 6404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-suc 4368  df-xp 4629  df-rel 4630  df-1o 6411
This theorem is referenced by:  djucomen  7209  djuassen  7210  xpdjuen  7211
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