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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 4029. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
epelg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3765 | . . . 4 | |
2 | elopab 3995 | . . . . . 6 | |
3 | vex 2560 | . . . . . . . . . . 11 | |
4 | vex 2560 | . . . . . . . . . . 11 | |
5 | 3, 4 | pm3.2i 257 | . . . . . . . . . 10 |
6 | opeqex 3986 | . . . . . . . . . 10 | |
7 | 5, 6 | mpbiri 157 | . . . . . . . . 9 |
8 | 7 | simpld 105 | . . . . . . . 8 |
9 | 8 | adantr 261 | . . . . . . 7 |
10 | 9 | exlimivv 1776 | . . . . . 6 |
11 | 2, 10 | sylbi 114 | . . . . 5 |
12 | df-eprel 4026 | . . . . 5 | |
13 | 11, 12 | eleq2s 2132 | . . . 4 |
14 | 1, 13 | sylbi 114 | . . 3 |
15 | 14 | a1i 9 | . 2 |
16 | elex 2566 | . . 3 | |
17 | 16 | a1i 9 | . 2 |
18 | eleq12 2102 | . . . 4 | |
19 | 18, 12 | brabga 4001 | . . 3 |
20 | 19 | expcom 109 | . 2 |
21 | 15, 17, 20 | pm5.21ndd 621 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cop 3378 class class class wbr 3764 copab 3817 cep 4024 |
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-v 2559 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-eprel 4026 |
This theorem is referenced by: epelc 4028 efrirr 4090 smoiso 5917 ecidg 6170 ordiso2 6357 ltpiord 6417 |
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