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Mirrors > Home > ILE Home > Th. List > xblm | Unicode version |
Description: A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xblm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elbl 14244 |
. . . 4
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2 | xmetge0 14218 |
. . . . . . . 8
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3 | 2 | 3expa 1204 |
. . . . . . 7
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4 | 3 | 3adantl3 1156 |
. . . . . 6
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5 | 0xr 8018 |
. . . . . . 7
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6 | xmetcl 14205 |
. . . . . . . . 9
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7 | 6 | 3expa 1204 |
. . . . . . . 8
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8 | 7 | 3adantl3 1156 |
. . . . . . 7
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9 | simpl3 1003 |
. . . . . . 7
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10 | xrlelttr 9820 |
. . . . . . 7
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11 | 5, 8, 9, 10 | mp3an2i 1352 |
. . . . . 6
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12 | 4, 11 | mpand 429 |
. . . . 5
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13 | 12 | expimpd 363 |
. . . 4
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14 | 1, 13 | sylbid 150 |
. . 3
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15 | 14 | exlimdv 1829 |
. 2
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16 | simpl2 1002 |
. . . 4
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17 | simpl1 1001 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | simpl3 1003 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | simpr 110 |
. . . . 5
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20 | xblcntr 14267 |
. . . . 5
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21 | 17, 16, 18, 19, 20 | syl112anc 1252 |
. . . 4
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22 | eleq1 2250 |
. . . . 5
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23 | 22 | spcegv 2837 |
. . . 4
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24 | 16, 21, 23 | sylc 62 |
. . 3
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25 | 24 | ex 115 |
. 2
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26 | 15, 25 | impbid 129 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-map 6664 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-2 8992 df-xadd 9787 df-psmet 13786 df-xmet 13787 df-bl 13789 |
This theorem is referenced by: blssioo 14398 |
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