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Theorem xpeq0r 4746
 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 4359 . . 3 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
2 0xp 4420 . . 3 (∅ × 𝐵) = ∅
31, 2syl6eq 2088 . 2 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
4 xpeq2 4360 . . 3 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5 xp0 4743 . . 3 (𝐴 × ∅) = ∅
64, 5syl6eq 2088 . 2 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
73, 6jaoi 636 1 ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 629   = wceq 1243  ∅c0 3224   × cxp 4343 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353 This theorem is referenced by: (None)
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