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Mirrors > Home > ILE Home > Th. List > blgt0 | Unicode version |
Description: A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
blgt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7812 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | simpl1 984 | . . 3 | |
4 | simpl2 985 | . . 3 | |
5 | elbl 12560 | . . . 4 | |
6 | 5 | simprbda 380 | . . 3 |
7 | xmetcl 12521 | . . 3 | |
8 | 3, 4, 6, 7 | syl3anc 1216 | . 2 |
9 | simpl3 986 | . 2 | |
10 | xmetge0 12534 | . . 3 | |
11 | 3, 4, 6, 10 | syl3anc 1216 | . 2 |
12 | 5 | simplbda 381 | . 2 |
13 | 2, 8, 9, 11, 12 | xrlelttrd 9593 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc0 7620 cxr 7799 clt 7800 cle 7801 cxmet 12149 cbl 12151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-2 8779 df-xadd 9560 df-psmet 12156 df-xmet 12157 df-bl 12159 |
This theorem is referenced by: (None) |
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