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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 3302 |
. . . 4
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2 | relxp 4656 |
. . . 4
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3 | relss 4634 |
. . . 4
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4 | 1, 2, 3 | mp2 16 |
. . 3
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5 | 4 | a1i 9 |
. 2
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6 | simpr 109 |
. . . . 5
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7 | brinxp2 4614 |
. . . . 5
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8 | 6, 7 | sylib 121 |
. . . 4
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9 | 8 | simp2d 995 |
. . 3
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10 | 8 | simp1d 994 |
. . 3
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11 | erinxp.r |
. . . . 5
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12 | 11 | adantr 274 |
. . . 4
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13 | 8 | simp3d 996 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 12, 13 | ersym 6449 |
. . 3
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15 | brinxp2 4614 |
. . 3
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16 | 9, 10, 14, 15 | syl3anbrc 1166 |
. 2
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17 | 10 | adantrr 471 |
. . 3
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18 | simprr 522 |
. . . . 5
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19 | brinxp2 4614 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 18, 19 | sylib 121 |
. . . 4
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21 | 20 | simp2d 995 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 11 | adantr 274 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 13 | adantrr 471 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 20 | simp3d 996 |
. . . 4
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25 | 22, 23, 24 | ertrd 6453 |
. . 3
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26 | brinxp2 4614 |
. . 3
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27 | 17, 21, 25, 26 | syl3anbrc 1166 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 11 | adantr 274 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | erinxp.a |
. . . . . . 7
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30 | 29 | sselda 3102 |
. . . . . 6
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31 | 28, 30 | erref 6457 |
. . . . 5
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32 | 31 | ex 114 |
. . . 4
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33 | 32 | pm4.71rd 392 |
. . 3
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34 | brin 3988 |
. . . 4
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35 | brxp 4578 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | anidm 394 |
. . . . . 6
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37 | 35, 36 | bitri 183 |
. . . . 5
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38 | 37 | anbi2i 453 |
. . . 4
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39 | 34, 38 | bitri 183 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 33, 39 | syl6bbr 197 |
. 2
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41 | 5, 16, 27, 40 | 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-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-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 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 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-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-er 6437 |
This theorem is referenced by: (None) |
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