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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 5357 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) = (∅ × (𝐶 ∩ 𝐷))) | |
2 | inxp 5488 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) | |
3 | 0xp 5435 | . . 3 ⊢ (∅ × (𝐶 ∩ 𝐷)) = ∅ | |
4 | 3 | eqcomi 2835 | . 2 ⊢ ∅ = (∅ × (𝐶 ∩ 𝐷)) |
5 | 1, 2, 4 | 3eqtr4g 2887 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∩ cin 3798 ∅c0 4145 × cxp 5341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4937 df-xp 5349 df-rel 5350 |
This theorem is referenced by: djudisj 5803 djuin 9058 xpdisjres 29959 esum2dlem 30700 nosupbnd2lem1 32401 noetalem2 32404 noetalem3 32405 bj-2upln1upl 33535 disjxp1 40056 |
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