Theorem epse 4264

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.

Step	Hyp	Ref	Expression
1		epel 4214	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 131	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2254	. . . . 5 |- x = { y | y _E x }
4		vex 2689	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2213	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3184	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4066	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2487	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4255	. 2 |- ( _E Se A <-> A. x e. A { y e. A | y _E x } e. _V )
10	8, 9	mpbir 145	1 |- _E Se A

Syntax hints:   e. wcel 1480   {cab 2125   A.wral 2416   {crab 2420   _Vcvv 2686   class class class wbr 3929   _E 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)