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Mirrors > Home > ILE Home > Th. List > xmeter | Unicode version |
Description: The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
xmeter.1 |
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Ref | Expression |
---|---|
xmeter |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmeter.1 |
. . . . 5
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2 | cnvimass 4987 |
. . . . 5
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3 | 1, 2 | eqsstri 3187 |
. . . 4
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4 | xmetf 13517 |
. . . 4
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5 | 3, 4 | fssdm 5376 |
. . 3
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6 | relxp 4732 |
. . 3
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7 | relss 4710 |
. . 3
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8 | 5, 6, 7 | mpisyl 1446 |
. 2
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9 | 1 | xmeterval 13602 |
. . . . 5
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10 | 9 | biimpa 296 |
. . . 4
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11 | 10 | simp2d 1010 |
. . 3
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12 | 10 | simp1d 1009 |
. . 3
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13 | simpl 109 |
. . . . 5
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14 | xmetsym 13535 |
. . . . 5
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15 | 13, 12, 11, 14 | syl3anc 1238 |
. . . 4
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16 | 10 | simp3d 1011 |
. . . 4
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17 | 15, 16 | eqeltrrd 2255 |
. . 3
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18 | 1 | xmeterval 13602 |
. . . 4
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19 | 18 | adantr 276 |
. . 3
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20 | 11, 12, 17, 19 | mpbir3and 1180 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 12 | adantrr 479 |
. . 3
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22 | 1 | xmeterval 13602 |
. . . . . 6
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23 | 22 | biimpa 296 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | adantrl 478 |
. . . 4
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25 | 24 | simp2d 1010 |
. . 3
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26 | simpl 109 |
. . . 4
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27 | 16 | adantrr 479 |
. . . . 5
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28 | 24 | simp3d 1011 |
. . . . 5
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29 | rexadd 9839 |
. . . . . 6
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30 | readdcl 7928 |
. . . . . 6
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31 | 29, 30 | eqeltrd 2254 |
. . . . 5
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32 | 27, 28, 31 | syl2anc 411 |
. . . 4
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33 | 11 | adantrr 479 |
. . . . 5
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34 | xmettri 13539 |
. . . . 5
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35 | 26, 21, 25, 33, 34 | syl13anc 1240 |
. . . 4
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36 | xmetlecl 13534 |
. . . 4
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37 | 26, 21, 25, 32, 35, 36 | syl122anc 1247 |
. . 3
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38 | 1 | xmeterval 13602 |
. . . 4
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39 | 38 | adantr 276 |
. . 3
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40 | 21, 25, 37, 39 | mpbir3and 1180 |
. 2
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41 | xmet0 13530 |
. . . . . . 7
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42 | 0re 7948 |
. . . . . . 7
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43 | 41, 42 | eqeltrdi 2268 |
. . . . . 6
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44 | 43 | ex 115 |
. . . . 5
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45 | 44 | pm4.71rd 394 |
. . . 4
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46 | df-3an 980 |
. . . . 5
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47 | anidm 396 |
. . . . . 6
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48 | 47 | anbi2ci 459 |
. . . . 5
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49 | 46, 48 | bitri 184 |
. . . 4
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50 | 45, 49 | bitr4di 198 |
. . 3
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51 | 1 | xmeterval 13602 |
. . 3
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52 | 50, 51 | bitr4d 191 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 8, 20, 40, 52 | iserd 6555 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-er 6529 df-map 6644 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-2 8967 df-xadd 9760 df-xmet 13155 |
This theorem is referenced by: blpnfctr 13606 xmetresbl 13607 |
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