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Mirrors > Home > ILE Home > Th. List > iserd | Unicode version |
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
iserd.1 | |
iserd.2 | |
iserd.3 | |
iserd.4 |
Ref | Expression |
---|---|
iserd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iserd.1 | . . 3 | |
2 | eqidd 2138 | . . 3 | |
3 | iserd.2 | . . . . . . . 8 | |
4 | 3 | ex 114 | . . . . . . 7 |
5 | iserd.3 | . . . . . . . 8 | |
6 | 5 | ex 114 | . . . . . . 7 |
7 | 4, 6 | jca 304 | . . . . . 6 |
8 | 7 | alrimiv 1846 | . . . . 5 |
9 | 8 | alrimiv 1846 | . . . 4 |
10 | 9 | alrimiv 1846 | . . 3 |
11 | dfer2 6423 | . . 3 | |
12 | 1, 2, 10, 11 | syl3anbrc 1165 | . 2 |
13 | 12 | adantr 274 | . . . . . . . 8 |
14 | simpr 109 | . . . . . . . 8 | |
15 | 13, 14 | erref 6442 | . . . . . . 7 |
16 | 15 | ex 114 | . . . . . 6 |
17 | vex 2684 | . . . . . . 7 | |
18 | 17, 17 | breldm 4738 | . . . . . 6 |
19 | 16, 18 | impbid1 141 | . . . . 5 |
20 | iserd.4 | . . . . 5 | |
21 | 19, 20 | bitr4d 190 | . . . 4 |
22 | 21 | eqrdv 2135 | . . 3 |
23 | ereq2 6430 | . . 3 | |
24 | 22, 23 | syl 14 | . 2 |
25 | 12, 24 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1329 wceq 1331 wcel 1480 class class class wbr 3924 cdm 4534 wrel 4539 wer 6419 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-er 6422 |
This theorem is referenced by: swoer 6450 eqer 6454 0er 6456 iinerm 6494 erinxp 6496 ecopover 6520 ecopoverg 6523 ener 6666 enq0er 7236 xmeter 12594 |
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