Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5129 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ∩ 𝐷) × (𝐴 ∩ 𝐵)) = ((𝐶 ∩ 𝐷) × ∅)) | |
2 | inxp 5254 | . 2 ⊢ ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ((𝐶 ∩ 𝐷) × (𝐴 ∩ 𝐵)) | |
3 | xp0 5552 | . . 3 ⊢ ((𝐶 ∩ 𝐷) × ∅) = ∅ | |
4 | 3 | eqcomi 2631 | . 2 ⊢ ∅ = ((𝐶 ∩ 𝐷) × ∅) |
5 | 1, 2, 4 | 3eqtr4g 2681 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∩ cin 3573 ∅c0 3915 × cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: xpsndisj 5557 |
Copyright terms: Public domain | W3C validator |