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Theorem xpeq0r 5069
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 4658 . . 3 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
2 0xp 4724 . . 3 (∅ × 𝐵) = ∅
31, 2eqtrdi 2238 . 2 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
4 xpeq2 4659 . . 3 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5 xp0 5066 . . 3 (𝐴 × ∅) = ∅
64, 5eqtrdi 2238 . 2 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
73, 6jaoi 717 1 ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709   = wceq 1364  c0 3437   × cxp 4642
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652
This theorem is referenced by:  sqxpeq0  5070
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