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Mirrors > Home > ILE Home > Th. List > epse | Unicode version |
Description: The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.) |
Ref | Expression |
---|---|
epse | Se |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epel 4214 | . . . . . . 7 | |
2 | 1 | bicomi 131 | . . . . . 6 |
3 | 2 | abbi2i 2254 | . . . . 5 |
4 | vex 2689 | . . . . 5 | |
5 | 3, 4 | eqeltrri 2213 | . . . 4 |
6 | rabssab 3184 | . . . 4 | |
7 | 5, 6 | ssexi 4066 | . . 3 |
8 | 7 | rgenw 2487 | . 2 |
9 | df-se 4255 | . 2 Se | |
10 | 8, 9 | mpbir 145 | 1 Se |
Colors of variables: wff set class |
Syntax hints: wcel 1480 cab 2125 wral 2416 crab 2420 cvv 2686 class class class wbr 3929 cep 4209 Se wse 4251 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-eprel 4211 df-se 4255 |
This theorem is referenced by: (None) |
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