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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 7530 | . 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 397 = wceq 1542 ∈ wcel 2107 ∅c0 4281 × cxp 5629 dom cdm 5631 (class class class)co 7350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-xp 5637 df-dm 5641 df-iota 6444 df-fv 6500 df-ov 7353 |
This theorem is referenced by: ndmovcl 7532 ndmovrcl 7533 ndmovcom 7534 ndmovass 7535 ndmovdistr 7536 om0x 8433 oaabs2 8563 omabs 8565 eceqoveq 8695 elpmi 8718 elmapex 8720 pmresg 8742 pmsspw 8749 addnidpi 10771 adderpq 10826 mulerpq 10827 elixx3g 13207 ndmioo 13221 elfz2 13361 fz0 13386 elfzoel1 13500 elfzoel2 13501 fzoval 13503 fzofi 13809 restsspw 17249 fucbas 17784 fuchom 17785 fuchomOLD 17786 xpcbas 18002 xpchomfval 18003 xpccofval 18006 restrcl 22436 ssrest 22455 resstopn 22465 iocpnfordt 22494 icomnfordt 22495 nghmfval 24014 isnghm 24015 topnfbey 29218 cvmtop1 33634 cvmtop2 33635 ndmico 43595 |
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