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| Mirrors > Home > MPE Home > Th. List > gt0ne0ii | Structured version Visualization version GIF version | ||
| Description: Positive implies nonzero. (Contributed by NM, 15-May-1999.) |
| Ref | Expression |
|---|---|
| lt2.1 | ⊢ 𝐴 ∈ ℝ |
| gt0ne0i.2 | ⊢ 0 < 𝐴 |
| Ref | Expression |
|---|---|
| gt0ne0ii | ⊢ 𝐴 ≠ 0 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | gt0ne0i.2 | . 2 ⊢ 0 < 𝐴 | |
| 2 | lt2.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
| 3 | 2 | gt0ne0i 11798 | . 2 ⊢ (0 < 𝐴 → 𝐴 ≠ 0) |
| 4 | 1, 3 | ax-mp 5 | 1 ⊢ 𝐴 ≠ 0 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ℝcr 11154 0cc0 11155 < clt 11295 |
| 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-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| 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-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-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: eqneg 11987 recgt0ii 12174 nnne0i 12306 2ne0 12370 3ne0 12372 4ne0 12374 8th4div3 12486 halfpm6th 12487 5recm6rec 12877 0.999... 15917 bpoly2 16093 bpoly3 16094 fsumcube 16096 efi4p 16173 resin4p 16174 recos4p 16175 ef01bndlem 16220 cos2bnd 16224 sincos2sgn 16230 ene0 16245 sinhalfpilem 26505 tan4thpi 26556 sincos6thpi 26558 sineq0 26566 coseq1 26567 efeq1 26570 cosne0 26571 efif1olem2 26585 efif1olem4 26587 eflogeq 26644 logf1o2 26692 cxpsqrt 26745 root1eq1 26798 sqrt2cxp2logb9e3 26842 ang180lem1 26852 ang180lem2 26853 ang180lem3 26854 2lgsoddprmlem1 27452 2lgsoddprmlem2 27453 chebbnd1lem3 27515 chebbnd1 27516 dp2cl 32862 dp2ltc 32869 dpfrac1 32874 dpmul4 32896 subfaclim 35193 bj-pinftynminfty 37228 taupilem1 37322 pine0 42348 acos1half 42388 proot1ex 43208 coseq0 45879 sinaover2ne0 45883 wallispi 46085 stirlinglem3 46091 stirlinglem15 46103 dirkertrigeqlem2 46114 dirkertrigeqlem3 46115 dirkertrigeq 46116 dirkeritg 46117 dirkercncflem1 46118 fourierdlem24 46146 fourierdlem95 46216 fourierswlem 46245 |
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