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| 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 5699 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) = (∅ × (𝐶 ∩ 𝐷))) | |
| 2 | inxp 5842 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) | |
| 3 | 0xp 5784 | . . 3 ⊢ (∅ × (𝐶 ∩ 𝐷)) = ∅ | |
| 4 | 3 | eqcomi 2746 | . 2 ⊢ ∅ = (∅ × (𝐶 ∩ 𝐷)) | 
| 5 | 1, 2, 4 | 3eqtr4g 2802 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∩ cin 3950 ∅c0 4333 × cxp 5683 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: djudisj 6187 xp01disjl 8530 djuin 9958 nosupbnd2lem1 27760 noetasuplem3 27780 noetasuplem4 27781 xpdisjres 32611 esum2dlem 34093 disjxp1 45074 | 
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