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| Mirrors > Home > MPE Home > Th. List > neg1ne0 | Structured version Visualization version GIF version | ||
| Description: -1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.) | 
| Ref | Expression | 
|---|---|
| neg1ne0 | ⊢ -1 ≠ 0 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-1cn 11213 | . 2 ⊢ 1 ∈ ℂ | |
| 2 | ax-1ne0 11224 | . 2 ⊢ 1 ≠ 0 | |
| 3 | 1, 2 | negne0i 11584 | 1 ⊢ -1 ≠ 0 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ≠ wne 2940 0cc0 11155 1c1 11156 -cneg 11493 | 
| 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-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 | 
| This theorem is referenced by: m1expcl2 14126 m1expeven 14150 iseraltlem2 15719 iseraltlem3 15720 iseralt 15721 m1expo 16412 m1exp1 16413 psgnunilem4 19515 m1expaddsub 19516 psgnuni 19517 cnmsgnsubg 21595 cnmsgngrp 21597 psgninv 21600 iblcnlem1 25823 itgcnlem 25825 dgrsub 26312 coseq00topi 26544 logtayl2 26704 root1eq1 26798 root1cj 26799 cxpeq 26800 angneg 26846 ang180lem1 26852 1cubrlem 26884 atantayl2 26981 basellem2 27125 isnsqf 27178 dchrfi 27299 dchrptlem1 27308 dchrptlem2 27309 lgsne0 27379 lgseisenlem1 27419 lgseisenlem2 27420 lgseisenlem4 27422 lgseisen 27423 lgsquadlem1 27424 lgsquad2lem1 27428 lgsquad3 27431 m1lgs 27432 hvsubcan 31093 hvsubcan2 31094 superpos 32373 sgnnbi 34548 signswch 34576 signstfvcl 34588 fwddifnp1 36166 proot1ex 43208 m1expevenALTV 47634 m1expoddALTV 47635 | 
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