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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 3783 | . 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅) | |
2 | uni0 3801 | . 2 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | eqtrdi 2206 | 1 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∅c0 3395 ∪ cuni 3774 × cxp 4586 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3396 df-sn 3567 df-uni 3775 |
This theorem is referenced by: (None) |
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