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Mirrors > Home > ILE Home > Th. List > unixp0im | GIF version |
Description: The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Ref | Expression |
---|---|
unixp0im | ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3798 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅) | |
2 | uni0 3816 | . 2 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | eqtrdi 2215 | 1 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1343 ∅c0 3409 ∪ cuni 3789 × cxp 4602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ral 2449 df-rex 2450 df-v 2728 df-dif 3118 df-in 3122 df-ss 3129 df-nul 3410 df-sn 3582 df-uni 3790 |
This theorem is referenced by: (None) |
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