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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 10442 | . 2 ⊢ (0 < 𝐴 → 𝐴 ≠ 0) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ 𝐴 ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ℝcr 9814 0cc0 9815 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: eqneg 10624 recgt0ii 10808 nnne0i 10932 2ne0 10990 3ne0 10992 4ne0 10994 8th4div3 11129 halfpm6th 11130 5recm6rec 11562 0.999... 14451 0.999...OLD 14452 bpoly2 14627 bpoly3 14628 fsumcube 14630 efi4p 14706 resin4p 14707 recos4p 14708 ef01bndlem 14753 cos2bnd 14757 sincos2sgn 14763 ene0 14776 sinhalfpilem 24019 sincos6thpi 24071 sineq0 24077 coseq1 24078 efeq1 24079 cosne0 24080 efif1olem2 24093 efif1olem4 24095 eflogeq 24152 logf1o2 24196 ecxp 24219 cxpsqrt 24249 root1eq1 24296 ang180lem1 24339 ang180lem2 24340 ang180lem3 24341 2lgsoddprmlem1 24933 2lgsoddprmlem2 24934 chebbnd1lem3 24960 chebbnd1 24961 subfaclim 30424 bj-pinftynminfty 32291 taupilem1 32344 proot1ex 36798 coseq0 38747 sinaover2ne0 38751 wallispi 38963 stirlinglem3 38969 stirlinglem15 38981 dirkertrigeqlem2 38992 dirkertrigeqlem3 38993 dirkertrigeq 38994 dirkeritg 38995 dirkercncflem1 38996 fourierdlem24 39024 fourierdlem95 39094 fourierswlem 39123 dp2cl 42309 dpfrac1 42312 dpfrac1OLD 42313 |
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