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

Step	Hyp	Ref	Expression
1		xpeq1 4359	. . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
2		0xp 4420	. . 3 ⊢ (∅ × 𝐵) = ∅
3	1, 2	syl6eq 2088	. 2 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
4		xpeq2 4360	. . 3 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5		xp0 4743	. . 3 ⊢ (𝐴 × ∅) = ∅
6	4, 5	syl6eq 2088	. 2 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
7	3, 6	jaoi 636	1 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)

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)