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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 7311 | . 2 ⊢ ((dom 𝐹 = (𝑆 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) | |
3 | 1, 2 | mpan 689 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∅c0 4243 × cxp 5517 dom cdm 5519 (class class class)co 7135 |
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-pow 5231 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-ral 3111 df-rex 3112 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-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: ndmovcl 7313 ndmovrcl 7314 ndmovcom 7315 ndmovass 7316 ndmovdistr 7317 om0x 8127 oaabs2 8255 omabs 8257 eceqoveq 8385 elpmi 8408 elmapex 8410 pmresg 8417 pmsspw 8424 addnidpi 10312 adderpq 10367 mulerpq 10368 elixx3g 12739 ndmioo 12753 elfz2 12892 fz0 12917 elfzoel1 13031 elfzoel2 13032 fzoval 13034 fzofi 13337 restsspw 16697 fucbas 17222 fuchom 17223 xpcbas 17420 xpchomfval 17421 xpccofval 17424 restrcl 21762 ssrest 21781 resstopn 21791 iocpnfordt 21820 icomnfordt 21821 nghmfval 23328 isnghm 23329 topnfbey 28254 cvmtop1 32620 cvmtop2 32621 ndmico 42203 |
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