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Mirrors > Home > ILE Home > Th. List > xpeq0r | GIF version |
Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.) |
Ref | Expression |
---|---|
xpeq0r | ⊢ ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 4658 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵)) | |
2 | 0xp 4724 | . . 3 ⊢ (∅ × 𝐵) = ∅ | |
3 | 1, 2 | eqtrdi 2238 | . 2 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅) |
4 | xpeq2 4659 | . . 3 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
5 | xp0 5066 | . . 3 ⊢ (𝐴 × ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2238 | . 2 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
7 | 3, 6 | jaoi 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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