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Mirrors > Home > ILE Home > Th. List > ercl | Unicode version |
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ersym.1 | |
ersym.2 |
Ref | Expression |
---|---|
ercl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ersym.1 | . . . 4 | |
2 | errel 6438 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | ersym.2 | . . 3 | |
5 | releldm 4774 | . . 3 | |
6 | 3, 4, 5 | syl2anc 408 | . 2 |
7 | erdm 6439 | . . 3 | |
8 | 1, 7 | syl 14 | . 2 |
9 | 6, 8 | eleqtrd 2218 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 class class class wbr 3929 cdm 4539 wrel 4544 wer 6426 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 df-er 6429 |
This theorem is referenced by: ercl2 6442 erthi 6475 qliftfun 6511 |
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