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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 4910 |
. . . . 5
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3 | 1, 2 | eqsstri 3134 |
. . . 4
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4 | xmetf 12558 |
. . . 4
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5 | 3, 4 | fssdm 5295 |
. . 3
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6 | relxp 4656 |
. . 3
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7 | relss 4634 |
. . 3
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8 | 5, 6, 7 | mpisyl 1423 |
. 2
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9 | 1 | xmeterval 12643 |
. . . . 5
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10 | 9 | biimpa 294 |
. . . 4
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11 | 10 | simp2d 995 |
. . 3
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12 | 10 | simp1d 994 |
. . 3
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13 | simpl 108 |
. . . . 5
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14 | xmetsym 12576 |
. . . . 5
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15 | 13, 12, 11, 14 | syl3anc 1217 |
. . . 4
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16 | 10 | simp3d 996 |
. . . 4
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17 | 15, 16 | eqeltrrd 2218 |
. . 3
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18 | 1 | xmeterval 12643 |
. . . 4
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19 | 18 | adantr 274 |
. . 3
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20 | 11, 12, 17, 19 | mpbir3and 1165 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 12 | adantrr 471 |
. . 3
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22 | 1 | xmeterval 12643 |
. . . . . 6
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23 | 22 | biimpa 294 |
. . . . 5
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24 | 23 | adantrl 470 |
. . . 4
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25 | 24 | simp2d 995 |
. . 3
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26 | simpl 108 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 16 | adantrr 471 |
. . . . 5
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28 | 24 | simp3d 996 |
. . . . 5
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29 | rexadd 9665 |
. . . . . 6
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30 | readdcl 7770 |
. . . . . 6
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31 | 29, 30 | eqeltrd 2217 |
. . . . 5
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32 | 27, 28, 31 | syl2anc 409 |
. . . 4
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33 | 11 | adantrr 471 |
. . . . 5
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34 | xmettri 12580 |
. . . . 5
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35 | 26, 21, 25, 33, 34 | syl13anc 1219 |
. . . 4
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36 | xmetlecl 12575 |
. . . 4
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37 | 26, 21, 25, 32, 35, 36 | syl122anc 1226 |
. . 3
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38 | 1 | xmeterval 12643 |
. . . 4
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39 | 38 | adantr 274 |
. . 3
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40 | 21, 25, 37, 39 | mpbir3and 1165 |
. 2
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41 | xmet0 12571 |
. . . . . . 7
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42 | 0re 7790 |
. . . . . . 7
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43 | 41, 42 | eqeltrdi 2231 |
. . . . . 6
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44 | 43 | ex 114 |
. . . . 5
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45 | 44 | pm4.71rd 392 |
. . . 4
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46 | df-3an 965 |
. . . . 5
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47 | anidm 394 |
. . . . . 6
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48 | 47 | anbi2ci 455 |
. . . . 5
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49 | 46, 48 | bitri 183 |
. . . 4
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50 | 45, 49 | syl6bbr 197 |
. . 3
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51 | 1 | xmeterval 12643 |
. . 3
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52 | 50, 51 | bitr4d 190 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 8, 20, 40, 52 | iserd 6463 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-er 6437 df-map 6552 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-2 8803 df-xadd 9590 df-xmet 12196 |
This theorem is referenced by: blpnfctr 12647 xmetresbl 12648 |
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