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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 11163 | . 2 ⊢ (0 < 𝐴 → 𝐴 ≠ 0) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ 𝐴 ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ℝcr 10524 0cc0 10525 < clt 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 |
This theorem is referenced by: eqneg 11348 recgt0ii 11534 nnne0i 11665 2ne0 11729 3ne0 11731 4ne0 11733 8th4div3 11845 halfpm6th 11846 5recm6rec 12230 0.999... 15225 bpoly2 15399 bpoly3 15400 fsumcube 15402 efi4p 15478 resin4p 15479 recos4p 15480 ef01bndlem 15525 cos2bnd 15529 sincos2sgn 15535 ene0 15550 sinhalfpilem 24976 sincos6thpi 25028 sineq0 25036 coseq1 25037 efeq1 25040 cosne0 25041 efif1olem2 25054 efif1olem4 25056 eflogeq 25112 logf1o2 25160 cxpsqrt 25213 root1eq1 25263 sqrt2cxp2logb9e3 25304 ang180lem1 25314 ang180lem2 25315 ang180lem3 25316 2lgsoddprmlem1 25911 2lgsoddprmlem2 25912 chebbnd1lem3 25974 chebbnd1 25975 dp2cl 30483 dp2ltc 30490 dpfrac1 30495 dpmul4 30517 subfaclim 32332 bj-pinftynminfty 34401 taupilem1 34484 proot1ex 39679 coseq0 42021 sinaover2ne0 42025 wallispi 42232 stirlinglem3 42238 stirlinglem15 42250 dirkertrigeqlem2 42261 dirkertrigeqlem3 42262 dirkertrigeq 42263 dirkeritg 42264 dirkercncflem1 42265 fourierdlem24 42293 fourierdlem95 42363 fourierswlem 42392 |
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