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Theorem epse 4064
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.)
Assertion
Ref Expression
epse E Se A

Proof of Theorem epse
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epel 4020 . . . . . . 7 (y E xy x)
21bicomi 123 . . . . . 6 (y xy E x)
32abbi2i 2149 . . . . 5 x = {yy E x}
4 vex 2554 . . . . 5 x V
53, 4eqeltrri 2108 . . . 4 {yy E x} V
6 rabssab 3021 . . . 4 {y Ay E x} ⊆ {yy E x}
75, 6ssexi 3886 . . 3 {y Ay E x} V
87rgenw 2370 . 2 x A {y Ay E x} V
9 df-se 4056 . 2 ( E Se Ax A {y Ay E x} V)
108, 9mpbir 134 1 E Se A
Colors of variables: wff set class
Syntax hints:   wcel 1390  {cab 2023  wral 2300  {crab 2304  Vcvv 2551   class class class wbr 3755   E cep 4015   Se wse 4055
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-eprel 4017  df-se 4056
This theorem is referenced by: (None)
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