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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 7320 | . 2 ⊢ ((dom 𝐹 = (𝑆 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) | |
3 | 1, 2 | mpan 686 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∅c0 4288 × cxp 5546 dom cdm 5548 (class class class)co 7145 |
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-pow 5257 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-ral 3140 df-rex 3141 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-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: ndmovcl 7322 ndmovrcl 7323 ndmovcom 7324 ndmovass 7325 ndmovdistr 7326 om0x 8133 oaabs2 8261 omabs 8263 eceqoveq 8391 elpmi 8414 elmapex 8416 pmresg 8423 pmsspw 8430 addnidpi 10311 adderpq 10366 mulerpq 10367 elixx3g 12739 ndmioo 12753 elfz2 12887 fz0 12910 elfzoel1 13024 elfzoel2 13025 fzoval 13027 fzofi 13330 restsspw 16693 fucbas 17218 fuchom 17219 xpcbas 17416 xpchomfval 17417 xpccofval 17420 restrcl 21693 ssrest 21712 resstopn 21722 iocpnfordt 21751 icomnfordt 21752 nghmfval 23258 isnghm 23259 topnfbey 28175 cvmtop1 32404 cvmtop2 32405 ndmico 41718 |
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