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Mirrors > Home > MPE Home > Th. List > metdsre | Unicode version |
Description: The distance from a point to a nonempty set in a proper metric space is a real number. (Contributed by Mario Carneiro, 5-Sep-2015.) |
Ref | Expression |
---|---|
metdscn.f |
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Ref | Expression |
---|---|
metdsre |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3605 |
. . 3
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2 | metxmet 18325 |
. . . . . . . . 9
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3 | metdscn.f |
. . . . . . . . . 10
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4 | 3 | metdsf 18839 |
. . . . . . . . 9
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5 | 2, 4 | sylan 458 |
. . . . . . . 8
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6 | 5 | adantr 452 |
. . . . . . 7
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7 | ffn 5558 |
. . . . . . 7
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8 | 6, 7 | syl 16 |
. . . . . 6
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9 | 5 | adantr 452 |
. . . . . . . . . . 11
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10 | simprr 734 |
. . . . . . . . . . 11
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11 | 9, 10 | ffvelrnd 5838 |
. . . . . . . . . 10
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12 | elxrge0 10972 |
. . . . . . . . . . 11
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13 | 12 | simplbi 447 |
. . . . . . . . . 10
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14 | 11, 13 | syl 16 |
. . . . . . . . 9
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15 | simpll 731 |
. . . . . . . . . 10
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16 | simpr 448 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | sselda 3316 |
. . . . . . . . . . 11
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18 | 17 | adantrr 698 |
. . . . . . . . . 10
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19 | metcl 18323 |
. . . . . . . . . 10
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20 | 15, 18, 10, 19 | syl3anc 1184 |
. . . . . . . . 9
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21 | 12 | simprbi 451 |
. . . . . . . . . 10
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22 | 11, 21 | syl 16 |
. . . . . . . . 9
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23 | 3 | metdsle 18843 |
. . . . . . . . . 10
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24 | 2, 23 | sylanl1 632 |
. . . . . . . . 9
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25 | xrrege0 10726 |
. . . . . . . . 9
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26 | 14, 20, 22, 24, 25 | syl22anc 1185 |
. . . . . . . 8
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27 | 26 | anassrs 630 |
. . . . . . 7
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28 | 27 | ralrimiva 2757 |
. . . . . 6
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29 | ffnfv 5861 |
. . . . . 6
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30 | 8, 28, 29 | sylanbrc 646 |
. . . . 5
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31 | 30 | ex 424 |
. . . 4
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32 | 31 | exlimdv 1643 |
. . 3
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33 | 1, 32 | syl5bi 209 |
. 2
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34 | 33 | 3impia 1150 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: metdscn2 18848 lebnumlem1 18947 lebnumlem3 18949 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 ax-cnex 9010 ax-resscn 9011 ax-1cn 9012 ax-icn 9013 ax-addcl 9014 ax-addrcl 9015 ax-mulcl 9016 ax-mulrcl 9017 ax-mulcom 9018 ax-addass 9019 ax-mulass 9020 ax-distr 9021 ax-i2m1 9022 ax-1ne0 9023 ax-1rid 9024 ax-rnegex 9025 ax-rrecex 9026 ax-cnre 9027 ax-pre-lttri 9028 ax-pre-lttrn 9029 ax-pre-ltadd 9030 ax-pre-mulgt0 9031 ax-pre-sup 9032 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-nel 2578 df-ral 2679 df-rex 2680 df-reu 2681 df-rmo 2682 df-rab 2683 df-v 2926 df-sbc 3130 df-csb 3220 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-op 3791 df-uni 3984 df-iun 4063 df-br 4181 df-opab 4235 df-mpt 4236 df-id 4466 df-po 4471 df-so 4472 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-f1 5426 df-fo 5427 df-f1o 5428 df-fv 5429 df-ov 6051 df-oprab 6052 df-mpt2 6053 df-1st 6316 df-2nd 6317 df-riota 6516 df-er 6872 df-ec 6874 df-map 6987 df-en 7077 df-dom 7078 df-sdom 7079 df-sup 7412 df-pnf 9086 df-mnf 9087 df-xr 9088 df-ltxr 9089 df-le 9090 df-sub 9257 df-neg 9258 df-div 9642 df-2 10022 df-rp 10577 df-xneg 10674 df-xadd 10675 df-xmul 10676 df-icc 10887 df-psmet 16657 df-xmet 16658 df-met 16659 df-bl 16660 |
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