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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 5052 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) = (∅ × (𝐶 ∩ 𝐷))) | |
2 | inxp 5176 | . 2 ⊢ ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ((𝐴 ∩ 𝐵) × (𝐶 ∩ 𝐷)) | |
3 | 0xp 5122 | . . 3 ⊢ (∅ × (𝐶 ∩ 𝐷)) = ∅ | |
4 | 3 | eqcomi 2619 | . 2 ⊢ ∅ = (∅ × (𝐶 ∩ 𝐷)) |
5 | 1, 2, 4 | 3eqtr4g 2669 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∩ cin 3539 ∅c0 3874 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: djudisj 5480 xpdisjres 28793 esum2dlem 29481 bj-2upln1upl 32205 disjxp1 38263 |
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