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Mirrors > Home > MPE Home > Th. List > ne0gt0d | Structured version Visualization version GIF version |
Description: A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ne0gt0d.2 | ⊢ (𝜑 → 0 ≤ 𝐴) |
ne0gt0d.3 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
ne0gt0d | ⊢ (𝜑 → 0 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0gt0d.3 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ne0gt0d.2 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) | |
4 | ne0gt0 10745 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≠ 0 ↔ 0 < 𝐴)) | |
5 | 2, 3, 4 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐴 ≠ 0 ↔ 0 < 𝐴)) |
6 | 1, 5 | mpbid 234 | 1 ⊢ (𝜑 → 0 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ℝcr 10536 0cc0 10537 < clt 10675 ≤ cle 10676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-addrcl 10598 ax-rnegex 10608 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 |
This theorem is referenced by: sqrtgt0 14618 absrpcl 14648 sqreulem 14719 fprodle 15350 efgt0 15456 abvgt0 19599 nmrpcl 23229 lebnumlem1 23565 ipcau2 23837 recxpcl 25258 mulcxp 25268 rlimcnp 25543 lgsdilem 25900 pntleml 26187 ttgcontlem1 26671 axsegconlem6 26708 axpaschlem 26726 axcontlem2 26751 axcontlem4 26753 axcontlem7 26756 xrge0iifhom 31180 cndprobprob 31696 usgrgt2cycl 32377 tan2h 34899 dvasin 34993 radcnvrat 40695 ioodvbdlimc1lem2 42266 ioodvbdlimc2lem 42268 fourierdlem30 42471 fourierdlem48 42488 fourierdlem49 42489 fourierdlem54 42494 fourierdlem102 42542 fourierdlem114 42554 sqwvfoura 42562 |
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