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Mirrors > Home > MPE Home > Th. List > 4ne0 | Structured version Visualization version GIF version |
Description: The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.) |
Ref | Expression |
---|---|
4ne0 | ⊢ 4 ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4re 11398 | . 2 ⊢ 4 ∈ ℝ | |
2 | 4pos 11427 | . 2 ⊢ 0 < 4 | |
3 | 1, 2 | gt0ne0ii 10856 | 1 ⊢ 4 ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2971 0cc0 10224 4c4 11370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-2 11376 df-3 11377 df-4 11378 |
This theorem is referenced by: 8th4div3 11540 div4p1lem1div2 11575 fldiv4p1lem1div2 12891 fldiv4lem1div2uz2 12892 fldiv4lem1div2 12893 discr 13255 sqoddm1div8 13284 4bc2eq6 13369 bpoly3 15125 bpoly4 15126 flodddiv4 15472 flodddiv4lt 15474 flodddiv4t2lthalf 15475 6lcm4e12 15664 cphipval2 23367 4cphipval2 23368 minveclem3 23539 uniioombl 23697 sincos4thpi 24607 sincos6thpi 24609 heron 24917 quad2 24918 dcubic 24925 mcubic 24926 cubic 24928 dquartlem1 24930 dquartlem2 24931 dquart 24932 quart1cl 24933 quart1lem 24934 quart1 24935 quartlem4 24939 quart 24940 log2tlbnd 25024 bclbnd 25357 bposlem7 25367 bposlem8 25368 bposlem9 25369 gausslemma2dlem0d 25436 gausslemma2dlem3 25445 gausslemma2dlem4 25446 gausslemma2dlem5 25448 m1lgs 25465 2lgslem1a2 25467 2lgslem1 25471 2lgslem2 25472 2lgslem3a 25473 2lgslem3b 25474 2lgslem3c 25475 2lgslem3d 25476 pntibndlem2 25632 4ipval2 28088 ipidsq 28090 dipcl 28092 dipcj 28094 dip0r 28097 dipcn 28100 ip1ilem 28206 ipasslem10 28219 polid2i 28539 lnopeq0i 29391 lnophmlem2 29401 quad3 32079 limclner 40627 stoweid 41023 wallispi2lem1 41031 stirlinglem3 41036 stirlinglem12 41045 stirlinglem13 41046 fouriersw 41191 |
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