Theorem epse 4097

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 4047	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 130	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2193	. . . . 5 |- x = { y | y _E x }
4		vex 2604	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2152	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3081	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 3916	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2418	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4088	. 2 |- ( _E Se A <-> A. x e. A { y e. A | y _E x } e. _V )
10	8, 9	mpbir 144	1 |- _E Se A

Syntax hints:   e. wcel 1433   {cab 2067   A.wral 2348   {crab 2352   _Vcvv 2601   class class class wbr 3785   _E cep 4042   Se wse 4084

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-eprel 4044  df-se 4088

This theorem is referenced by: (None)