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Mirrors > Home > NFE Home > Th. List > ersymtr | Unicode version |
Description: Equivalence relationship as symmetric, transitive relationship. (Contributed by SF, 22-Feb-2015.) |
Ref | Expression |
---|---|
ersymtr | Er Sym Trans |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-er 5909 | . . 3 Er Sym Trans | |
2 | 1 | breqi 4645 | . 2 Er Sym Trans |
3 | brin 4693 | . 2 Sym Trans Sym Trans | |
4 | 2, 3 | bitri 240 | 1 Er Sym Trans |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 cin 3208 class class class wbr 4639 Trans ctrans 5888 Sym csym 5897 Er cer 5898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-br 4640 df-er 5909 |
This theorem is referenced by: iserd 5942 ersym 5952 ertr 5954 ertrd 5955 |
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