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Theorem xpeq0r 4811
Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
xpeq0r  |-  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  X.  B )  =  (/) )

Proof of Theorem xpeq0r
StepHypRef Expression
1 xpeq1 4418 . . 3  |-  ( A  =  (/)  ->  ( A  X.  B )  =  ( (/)  X.  B
) )
2 0xp 4479 . . 3  |-  ( (/)  X.  B )  =  (/)
31, 2syl6eq 2133 . 2  |-  ( A  =  (/)  ->  ( A  X.  B )  =  (/) )
4 xpeq2 4419 . . 3  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
5 xp0 4808 . . 3  |-  ( A  X.  (/) )  =  (/)
64, 5syl6eq 2133 . 2  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
73, 6jaoi 669 1  |-  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  X.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1287   (/)c0 3272    X. cxp 4402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-rel 4411  df-cnv 4412
This theorem is referenced by: (None)
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