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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 4288. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
epelg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4001 |
. . . 4
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2 | elopab 4254 |
. . . . . 6
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3 | vex 2740 |
. . . . . . . . . . 11
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4 | vex 2740 |
. . . . . . . . . . 11
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5 | 3, 4 | pm3.2i 272 |
. . . . . . . . . 10
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6 | opeqex 4245 |
. . . . . . . . . 10
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7 | 5, 6 | mpbiri 168 |
. . . . . . . . 9
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8 | 7 | simpld 112 |
. . . . . . . 8
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9 | 8 | adantr 276 |
. . . . . . 7
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10 | 9 | exlimivv 1896 |
. . . . . 6
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11 | 2, 10 | sylbi 121 |
. . . . 5
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12 | df-eprel 4285 |
. . . . 5
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13 | 11, 12 | eleq2s 2272 |
. . . 4
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14 | 1, 13 | sylbi 121 |
. . 3
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15 | 14 | a1i 9 |
. 2
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16 | elex 2748 |
. . 3
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17 | 16 | a1i 9 |
. 2
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18 | eleq12 2242 |
. . . 4
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19 | 18, 12 | brabga 4260 |
. . 3
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20 | 19 | expcom 116 |
. 2
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21 | 15, 17, 20 | pm5.21ndd 705 |
1
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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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-eprel 4285 |
This theorem is referenced by: epelc 4287 efrirr 4349 smoiso 6296 ecidg 6592 ordiso2 7027 ltpiord 7296 |
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