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Mirrors > Home > ILE Home > Th. List > erref | Unicode version |
Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ersymb.1 |
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erref.2 |
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Ref | Expression |
---|---|
erref |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erref.2 |
. . . 4
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2 | ersymb.1 |
. . . . 5
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3 | erdm 6568 |
. . . . 5
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4 | 2, 3 | syl 14 |
. . . 4
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5 | 1, 4 | eleqtrrd 2269 |
. . 3
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6 | eldmg 4840 |
. . . 4
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7 | 1, 6 | syl 14 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 5, 7 | mpbid 147 |
. 2
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9 | 2 | adantr 276 |
. . 3
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10 | simpr 110 |
. . 3
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11 | 9, 10, 10 | ertr4d 6577 |
. 2
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12 | 8, 11 | exlimddv 1910 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-er 6558 |
This theorem is referenced by: iserd 6584 erth 6604 iinerm 6632 erinxp 6634 qusgrp 13168 |
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