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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 | 4nn 12350 | . 2 ⊢ 4 ∈ ℕ | |
| 2 | 1 | nnne0i 12307 | 1 ⊢ 4 ≠ 0 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ≠ wne 2939 0cc0 11156 4c4 12324 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 | 
| This theorem is referenced by: 8th4div3 12488 div4p1lem1div2 12523 fldiv4p1lem1div2 13876 fldiv4lem1div2uz2 13877 fldiv4lem1div2 13878 discr 14280 sqoddm1div8 14283 4bc2eq6 14369 bpoly3 16095 bpoly4 16096 flodddiv4 16453 flodddiv4lt 16455 flodddiv4t2lthalf 16456 6lcm4e12 16654 cphipval2 25276 4cphipval2 25277 minveclem3 25464 uniioombl 25625 sincos4thpi 26556 tan4thpi 26557 sincos6thpi 26559 heron 26882 quad2 26883 dcubic 26890 mcubic 26891 cubic 26893 dquartlem1 26895 dquartlem2 26896 dquart 26897 quart1cl 26898 quart1lem 26899 quart1 26900 quartlem4 26904 quart 26905 log2tlbnd 26989 bclbnd 27325 bposlem7 27335 bposlem8 27336 bposlem9 27337 gausslemma2dlem0d 27404 gausslemma2dlem3 27413 gausslemma2dlem4 27414 gausslemma2dlem5 27416 m1lgs 27433 2lgslem1a2 27435 2lgslem1 27439 2lgslem2 27440 2lgslem3a 27441 2lgslem3b 27442 2lgslem3c 27443 2lgslem3d 27444 pntibndlem2 27636 4ipval2 30728 ipidsq 30730 dipcl 30732 dipcj 30734 dip0r 30737 dipcn 30740 ip1ilem 30846 ipasslem10 30859 polid2i 31177 lnopeq0i 32027 lnophmlem2 32037 quad3d 32755 quad3 35676 flt4lem5e 42671 limclner 45671 stoweid 46083 wallispi2lem1 46091 stirlinglem3 46096 stirlinglem12 46105 stirlinglem13 46106 fouriersw 46251 usgrexmpl2lem 47990 usgrexmpl2nb4 47999 | 
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