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Mirrors > Home > ILE Home > Th. List > epelg | Unicode version |
Description: The epsilon relation and membership are the same. General version of epel 4264. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
epelg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3977 | . . . 4 | |
2 | elopab 4230 | . . . . . 6 | |
3 | vex 2724 | . . . . . . . . . . 11 | |
4 | vex 2724 | . . . . . . . . . . 11 | |
5 | 3, 4 | pm3.2i 270 | . . . . . . . . . 10 |
6 | opeqex 4221 | . . . . . . . . . 10 | |
7 | 5, 6 | mpbiri 167 | . . . . . . . . 9 |
8 | 7 | simpld 111 | . . . . . . . 8 |
9 | 8 | adantr 274 | . . . . . . 7 |
10 | 9 | exlimivv 1883 | . . . . . 6 |
11 | 2, 10 | sylbi 120 | . . . . 5 |
12 | df-eprel 4261 | . . . . 5 | |
13 | 11, 12 | eleq2s 2259 | . . . 4 |
14 | 1, 13 | sylbi 120 | . . 3 |
15 | 14 | a1i 9 | . 2 |
16 | elex 2732 | . . 3 | |
17 | 16 | a1i 9 | . 2 |
18 | eleq12 2229 | . . . 4 | |
19 | 18, 12 | brabga 4236 | . . 3 |
20 | 19 | expcom 115 | . 2 |
21 | 15, 17, 20 | pm5.21ndd 695 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2721 cop 3573 class class class wbr 3976 copab 4036 cep 4259 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-eprel 4261 |
This theorem is referenced by: epelc 4263 efrirr 4325 smoiso 6261 ecidg 6556 ordiso2 6991 ltpiord 7251 |
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