Theorem xpeq0r 5205

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 4783	. . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
2		0xp 4850	. . 3 ⊢ (∅ × 𝐵) = ∅
3	1, 2	eqtrdi 2287	. 2 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
4		xpeq2 4784	. . 3 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5		xp0 5202	. . 3 ⊢ (𝐴 × ∅) = ∅
6	4, 5	eqtrdi 2287	. 2 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
7	3, 6	jaoi 728	1 ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)

Syntax hints:   → wi 4   ∨ wo 720   = wceq 1402  ∅c0 3520   × cxp 4767

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777

This theorem is referenced by:  sqxpeq0  5206