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Mirrors > Home > ILE Home > Th. List > ercnv | Unicode version |
Description: The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ercnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | errel 6518 | . 2 | |
2 | relcnv 4987 | . . 3 | |
3 | id 19 | . . . . . 6 | |
4 | 3 | ersymb 6523 | . . . . 5 |
5 | vex 2733 | . . . . . . 7 | |
6 | vex 2733 | . . . . . . 7 | |
7 | 5, 6 | brcnv 4792 | . . . . . 6 |
8 | df-br 3988 | . . . . . 6 | |
9 | 7, 8 | bitr3i 185 | . . . . 5 |
10 | df-br 3988 | . . . . 5 | |
11 | 4, 9, 10 | 3bitr3g 221 | . . . 4 |
12 | 11 | eqrelrdv2 4708 | . . 3 |
13 | 2, 12 | mpanl1 432 | . 2 |
14 | 1, 13 | mpancom 420 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 cop 3584 class class class wbr 3987 ccnv 4608 wrel 4614 wer 6506 |
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 4105 ax-pow 4158 ax-pr 4192 |
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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-er 6509 |
This theorem is referenced by: errn 6531 |
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