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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 5003 |
. . . . 5
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3 | 1, 2 | eqsstri 3199 |
. . . 4
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4 | xmetf 14146 |
. . . 4
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5 | 3, 4 | fssdm 5392 |
. . 3
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6 | relxp 4747 |
. . 3
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7 | relss 4725 |
. . 3
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8 | 5, 6, 7 | mpisyl 1456 |
. 2
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9 | 1 | xmeterval 14231 |
. . . . 5
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10 | 9 | biimpa 296 |
. . . 4
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11 | 10 | simp2d 1011 |
. . 3
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12 | 10 | simp1d 1010 |
. . 3
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13 | simpl 109 |
. . . . 5
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14 | xmetsym 14164 |
. . . . 5
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15 | 13, 12, 11, 14 | syl3anc 1248 |
. . . 4
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16 | 10 | simp3d 1012 |
. . . 4
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17 | 15, 16 | eqeltrrd 2265 |
. . 3
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18 | 1 | xmeterval 14231 |
. . . 4
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19 | 18 | adantr 276 |
. . 3
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20 | 11, 12, 17, 19 | mpbir3and 1181 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 12 | adantrr 479 |
. . 3
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22 | 1 | xmeterval 14231 |
. . . . . 6
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23 | 22 | biimpa 296 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | adantrl 478 |
. . . 4
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25 | 24 | simp2d 1011 |
. . 3
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26 | simpl 109 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 16 | adantrr 479 |
. . . . 5
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28 | 24 | simp3d 1012 |
. . . . 5
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29 | rexadd 9866 |
. . . . . 6
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30 | readdcl 7951 |
. . . . . 6
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31 | 29, 30 | eqeltrd 2264 |
. . . . 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 14168 |
. . . . 5
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35 | 26, 21, 25, 33, 34 | syl13anc 1250 |
. . . 4
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36 | xmetlecl 14163 |
. . . 4
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37 | 26, 21, 25, 32, 35, 36 | syl122anc 1257 |
. . 3
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38 | 1 | xmeterval 14231 |
. . . 4
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39 | 38 | adantr 276 |
. . 3
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40 | 21, 25, 37, 39 | mpbir3and 1181 |
. 2
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41 | xmet0 14159 |
. . . . . . 7
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42 | 0re 7971 |
. . . . . . 7
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43 | 41, 42 | eqeltrdi 2278 |
. . . . . 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 981 |
. . . . 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 14231 |
. . 3
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52 | 50, 51 | bitr4d 191 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 8, 20, 40, 52 | iserd 6575 |
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-apti 7940 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-er 6549 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-xmet 13730 |
This theorem is referenced by: blpnfctr 14235 xmetresbl 14236 |
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