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Mirrors > Home > MPE Home > Th. List > xpsndisj | Structured version Visualization version GIF version |
Description: Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.) |
Ref | Expression |
---|---|
xpsndisj | ⊢ (𝐵 ≠ 𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn2 4608 | . 2 ⊢ (𝐵 ≠ 𝐷 → ({𝐵} ∩ {𝐷}) = ∅) | |
2 | xpdisj2 5986 | . 2 ⊢ (({𝐵} ∩ {𝐷}) = ∅ → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ≠ wne 2987 ∩ cin 3880 ∅c0 4243 {csn 4525 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 |
This theorem is referenced by: xp01disj 8103 unxpdom2 8710 sucxpdom 8711 |
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