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Mirrors > Home > MPE Home > Th. List > xpdisj1 | Structured version Visualization version GIF version |
Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
Ref | Expression |
---|---|
xpdisj1 | ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 5703 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) = (∅ × (𝐶 ∩ 𝐷))) | |
2 | inxp 5845 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) | |
3 | 0xp 5787 | . . 3 ⊢ (∅ × (𝐶 ∩ 𝐷)) = ∅ | |
4 | 3 | eqcomi 2744 | . 2 ⊢ ∅ = (∅ × (𝐶 ∩ 𝐷)) |
5 | 1, 2, 4 | 3eqtr4g 2800 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3962 ∅c0 4339 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: djudisj 6189 xp01disjl 8529 djuin 9956 nosupbnd2lem1 27775 noetasuplem3 27795 noetasuplem4 27796 xpdisjres 32618 esum2dlem 34073 disjxp1 45009 |
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