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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 4143. (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 3868 |
. . . 4
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2 | elopab 4109 |
. . . . . 6
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3 | vex 2636 |
. . . . . . . . . . 11
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4 | vex 2636 |
. . . . . . . . . . 11
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5 | 3, 4 | pm3.2i 267 |
. . . . . . . . . 10
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6 | opeqex 4100 |
. . . . . . . . . 10
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7 | 5, 6 | mpbiri 167 |
. . . . . . . . 9
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8 | 7 | simpld 111 |
. . . . . . . 8
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9 | 8 | adantr 271 |
. . . . . . 7
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10 | 9 | exlimivv 1831 |
. . . . . 6
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11 | 2, 10 | sylbi 120 |
. . . . 5
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12 | df-eprel 4140 |
. . . . 5
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13 | 11, 12 | eleq2s 2189 |
. . . 4
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14 | 1, 13 | sylbi 120 |
. . 3
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15 | 14 | a1i 9 |
. 2
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16 | elex 2644 |
. . 3
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17 | 16 | a1i 9 |
. 2
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18 | eleq12 2159 |
. . . 4
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19 | 18, 12 | brabga 4115 |
. . 3
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20 | 19 | expcom 115 |
. 2
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21 | 15, 17, 20 | pm5.21ndd 659 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-br 3868 df-opab 3922 df-eprel 4140 |
This theorem is referenced by: epelc 4142 efrirr 4204 smoiso 6105 ecidg 6396 ordiso2 6808 ltpiord 6975 |
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