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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 4984 |
. . . . 5
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3 | 1, 2 | eqsstri 3185 |
. . . 4
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4 | xmetf 13430 |
. . . 4
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5 | 3, 4 | fssdm 5372 |
. . 3
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6 | relxp 4729 |
. . 3
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7 | relss 4707 |
. . 3
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8 | 5, 6, 7 | mpisyl 1444 |
. 2
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9 | 1 | xmeterval 13515 |
. . . . 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 13448 |
. . . . 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 2253 |
. . 3
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18 | 1 | xmeterval 13515 |
. . . 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 13515 |
. . . . . 6
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23 | 22 | biimpa 296 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | adantrl 478 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | simp2d 1010 |
. . 3
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26 | simpl 109 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 16 | adantrr 479 |
. . . . 5
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28 | 24 | simp3d 1011 |
. . . . 5
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29 | rexadd 9823 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | readdcl 7912 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | eqeltrd 2252 |
. . . . 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 13452 |
. . . . 5
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35 | 26, 21, 25, 33, 34 | syl13anc 1240 |
. . . 4
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36 | xmetlecl 13447 |
. . . 4
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37 | 26, 21, 25, 32, 35, 36 | syl122anc 1247 |
. . 3
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38 | 1 | xmeterval 13515 |
. . . 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 13443 |
. . . . . . 7
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42 | 0re 7932 |
. . . . . . 7
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43 | 41, 42 | eqeltrdi 2266 |
. . . . . 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
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
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 13515 |
. . 3
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52 | 50, 51 | bitr4d 191 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 8, 20, 40, 52 | iserd 6551 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 |
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 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-po 4290 df-iso 4291 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-er 6525 df-map 6640 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-2 8951 df-xadd 9744 df-xmet 13068 |
This theorem is referenced by: blpnfctr 13519 xmetresbl 13520 |
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