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Mirrors > Home > ILE Home > Th. List > erthi | Unicode version |
Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
erthi.1 | |
erthi.2 |
Ref | Expression |
---|---|
erthi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erthi.2 | . 2 | |
2 | erthi.1 | . . 3 | |
3 | 2, 1 | ercl 6408 | . . 3 |
4 | 2, 3 | erth 6441 | . 2 |
5 | 1, 4 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 class class class wbr 3899 wer 6394 cec 6395 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-er 6397 df-ec 6399 |
This theorem is referenced by: qsel 6474 th3qlem1 6499 mulcanenqec 7162 mulcanenq0ec 7221 addnq0mo 7223 mulnq0mo 7224 addsrmo 7519 mulsrmo 7520 blpnfctr 12535 |
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