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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 4247 | . 2 ⊢ (𝐵 ≠ 𝐷 → ({𝐵} ∩ {𝐷}) = ∅) | |
2 | xpdisj2 5556 | . 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 1483 ≠ wne 2794 ∩ cin 3573 ∅c0 3915 {csn 4177 × 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-ne 2795 df-ral 2917 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: xp01disj 7576 unxpdom2 8168 sucxpdom 8169 |
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