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Theorem xpeq0r 4673
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 × B) = ∅)

Proof of Theorem xpeq0r
StepHypRef Expression
1 xpeq1 4286 . . 3 (A = ∅ → (A × B) = (∅ × B))
2 0xp 4347 . . 3 (∅ × B) = ∅
31, 2syl6eq 2070 . 2 (A = ∅ → (A × B) = ∅)
4 xpeq2 4287 . . 3 (B = ∅ → (A × B) = (A × ∅))
5 xp0 4670 . . 3 (A × ∅) = ∅
64, 5syl6eq 2070 . 2 (B = ∅ → (A × B) = ∅)
73, 6jaoi 623 1 ((A = ∅ B = ∅) → (A × B) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228  c0 3201   × cxp 4270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280
This theorem is referenced by: (None)
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