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Mirrors > Home > MPE Home > Th. List > xpdisj2 | Structured version Visualization version GIF version |
Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
Ref | Expression |
---|---|
xpdisj2 | ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq2 5569 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ∩ 𝐷) × (𝐴 ∩ 𝐵)) = ((𝐶 ∩ 𝐷) × ∅)) | |
2 | inxp 5696 | . 2 ⊢ ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ((𝐶 ∩ 𝐷) × (𝐴 ∩ 𝐵)) | |
3 | xp0 6008 | . . 3 ⊢ ((𝐶 ∩ 𝐷) × ∅) = ∅ | |
4 | 3 | eqcomi 2827 | . 2 ⊢ ∅ = ((𝐶 ∩ 𝐷) × ∅) |
5 | 1, 2, 4 | 3eqtr4g 2878 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∩ cin 3932 ∅c0 4288 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 |
This theorem is referenced by: xpsndisj 6013 |
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