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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 | |
erref.2 |
Ref | Expression |
---|---|
erref |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erref.2 | . . . 4 | |
2 | ersymb.1 | . . . . 5 | |
3 | erdm 6523 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 1, 4 | eleqtrrd 2250 | . . 3 |
6 | eldmg 4806 | . . . 4 | |
7 | 1, 6 | syl 14 | . . 3 |
8 | 5, 7 | mpbid 146 | . 2 |
9 | 2 | adantr 274 | . . 3 |
10 | simpr 109 | . . 3 | |
11 | 9, 10, 10 | ertr4d 6532 | . 2 |
12 | 8, 11 | exlimddv 1891 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 class class class wbr 3989 cdm 4611 wer 6510 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-er 6513 |
This theorem is referenced by: iserd 6539 erth 6557 iinerm 6585 erinxp 6587 |
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