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Mirrors > Home > ILE Home > Th. List > erth2 | Unicode version |
Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) |
Ref | Expression |
---|---|
erth2.1 | |
erth2.2 |
Ref | Expression |
---|---|
erth2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erth2.1 | . . 3 | |
2 | 1 | ersymb 6120 | . 2 |
3 | erth2.2 | . . . 4 | |
4 | 1, 3 | erth 6150 | . . 3 |
5 | eqcom 2042 | . . 3 | |
6 | 4, 5 | syl6bb 185 | . 2 |
7 | 2, 6 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 98 wceq 1243 wcel 1393 class class class wbr 3764 wer 6103 cec 6104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-er 6106 df-ec 6108 |
This theorem is referenced by: qliftel 6186 |
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