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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 4327. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
| Ref | Expression |
|---|---|
| epelg |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 4034 |
. . . 4
| |
| 2 | elopab 4292 |
. . . . . 6
| |
| 3 | vex 2766 |
. . . . . . . . . . 11
| |
| 4 | vex 2766 |
. . . . . . . . . . 11
| |
| 5 | 3, 4 | pm3.2i 272 |
. . . . . . . . . 10
|
| 6 | opeqex 4282 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | mpbiri 168 |
. . . . . . . . 9
|
| 8 | 7 | simpld 112 |
. . . . . . . 8
|
| 9 | 8 | adantr 276 |
. . . . . . 7
|
| 10 | 9 | exlimivv 1911 |
. . . . . 6
|
| 11 | 2, 10 | sylbi 121 |
. . . . 5
|
| 12 | df-eprel 4324 |
. . . . 5
| |
| 13 | 11, 12 | eleq2s 2291 |
. . . 4
|
| 14 | 1, 13 | sylbi 121 |
. . 3
|
| 15 | 14 | a1i 9 |
. 2
|
| 16 | elex 2774 |
. . 3
| |
| 17 | 16 | a1i 9 |
. 2
|
| 18 | eleq12 2261 |
. . . 4
| |
| 19 | 18, 12 | brabga 4298 |
. . 3
|
| 20 | 19 | expcom 116 |
. 2
|
| 21 | 15, 17, 20 | pm5.21ndd 706 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-br 4034 df-opab 4095 df-eprel 4324 |
| This theorem is referenced by: epelc 4326 efrirr 4388 smoiso 6360 ecidg 6658 ordiso2 7101 ltpiord 7386 |
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