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| Mirrors > Home > MPE Home > Th. List > ndmov | Structured version Visualization version GIF version | ||
| Description: The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.) |
| Ref | Expression |
|---|---|
| ndmov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
| Ref | Expression |
|---|---|
| ndmov | ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ndmov.1 | . 2 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
| 2 | ndmovg 7616 | . 2 ⊢ ((dom 𝐹 = (𝑆 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 × cxp 5683 dom cdm 5685 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: ndmovcl 7618 ndmovrcl 7619 ndmovcom 7620 ndmovass 7621 ndmovdistr 7622 om0x 8557 oaabs2 8687 omabs 8689 eceqoveq 8862 elpmi 8886 elmapex 8888 pmresg 8910 pmsspw 8917 addnidpi 10941 adderpq 10996 mulerpq 10997 elixx3g 13400 ndmioo 13414 elfz2 13554 fz0 13579 elfzoel1 13697 elfzoel2 13698 fzoval 13700 fzofi 14015 restsspw 17476 fucbas 18008 fuchom 18009 xpcbas 18223 xpchomfval 18224 xpccofval 18227 restrcl 23165 ssrest 23184 resstopn 23194 iocpnfordt 23223 icomnfordt 23224 nghmfval 24743 isnghm 24744 topnfbey 30488 cvmtop1 35265 cvmtop2 35266 ndmico 45579 |
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