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Mirrors > Home > ILE Home > Th. List > erinxp | Unicode version |
Description: A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
erinxp.r |
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erinxp.a |
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Ref | Expression |
---|---|
erinxp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3152 |
. . . 4
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2 | relxp 4390 |
. . . 4
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3 | relss 4370 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 1, 2, 3 | mp2 16 |
. . 3
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5 | 4 | a1i 9 |
. 2
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6 | simpr 103 |
. . . . 5
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7 | brinxp2 4350 |
. . . . 5
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8 | 6, 7 | sylib 127 |
. . . 4
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9 | 8 | simp2d 916 |
. . 3
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10 | 8 | simp1d 915 |
. . 3
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11 | erinxp.r |
. . . . 5
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12 | 11 | adantr 261 |
. . . 4
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13 | 8 | simp3d 917 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 12, 13 | ersym 6054 |
. . 3
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15 | brinxp2 4350 |
. . 3
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16 | 9, 10, 14, 15 | syl3anbrc 1087 |
. 2
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17 | 10 | adantrr 448 |
. . 3
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18 | simprr 484 |
. . . . 5
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19 | brinxp2 4350 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 18, 19 | sylib 127 |
. . . 4
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21 | 20 | simp2d 916 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 11 | adantr 261 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 13 | adantrr 448 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 20 | simp3d 917 |
. . . 4
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25 | 22, 23, 24 | ertrd 6058 |
. . 3
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26 | brinxp2 4350 |
. . 3
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27 | 17, 21, 25, 26 | syl3anbrc 1087 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 11 | adantr 261 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | erinxp.a |
. . . . . . 7
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30 | 29 | sselda 2939 |
. . . . . 6
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31 | 28, 30 | erref 6062 |
. . . . 5
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32 | 31 | ex 108 |
. . . 4
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33 | 32 | pm4.71rd 374 |
. . 3
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34 | brin 3802 |
. . . 4
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35 | brxp 4318 |
. . . . . 6
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36 | anidm 376 |
. . . . . 6
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37 | 35, 36 | bitri 173 |
. . . . 5
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38 | 37 | anbi2i 430 |
. . . 4
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39 | 34, 38 | bitri 173 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 33, 39 | syl6bbr 187 |
. 2
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41 | 5, 16, 27, 40 | iserd 6068 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-er 6042 |
This theorem is referenced by: (None) |
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