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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 5569 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) = (∅ × (𝐶 ∩ 𝐷))) | |
2 | inxp 5703 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) | |
3 | 0xp 5649 | . . 3 ⊢ (∅ × (𝐶 ∩ 𝐷)) = ∅ | |
4 | 3 | eqcomi 2830 | . 2 ⊢ ∅ = (∅ × (𝐶 ∩ 𝐷)) |
5 | 1, 2, 4 | 3eqtr4g 2881 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∩ cin 3935 ∅c0 4291 × cxp 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-xp 5561 df-rel 5562 |
This theorem is referenced by: djudisj 6024 xp01disjl 8121 djuin 9347 xpdisjres 30348 esum2dlem 31351 nosupbnd2lem1 33215 noetalem2 33218 noetalem3 33219 disjxp1 41351 |
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