Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10747 | . . 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 3018 class class class wbr 5068 ℝcr 10538 0cc0 10539 < clt 10677 ≤ cle 10678 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-addrcl 10600 ax-rnegex 10610 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: sqrtgt0 14620 absrpcl 14650 sqreulem 14721 fprodle 15352 efgt0 15458 abvgt0 19601 nmrpcl 23231 lebnumlem1 23567 ipcau2 23839 recxpcl 25260 mulcxp 25270 rlimcnp 25545 lgsdilem 25902 pntleml 26189 ttgcontlem1 26673 axsegconlem6 26710 axpaschlem 26728 axcontlem2 26753 axcontlem4 26755 axcontlem7 26758 xrge0iifhom 31182 cndprobprob 31698 usgrgt2cycl 32379 tan2h 34886 dvasin 34980 radcnvrat 40653 ioodvbdlimc1lem2 42224 ioodvbdlimc2lem 42226 fourierdlem30 42429 fourierdlem48 42446 fourierdlem49 42447 fourierdlem54 42452 fourierdlem102 42500 fourierdlem114 42512 sqwvfoura 42520 |
Copyright terms: Public domain | W3C validator |