Theorem epse 4222

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 4172	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 131	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2227	. . . . 5 |- x = { y | y _E x }
4		vex 2658	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2186	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3148	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4024	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2459	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4213	. 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 1461   {cab 2099   A.wral 2388   {crab 2392   _Vcvv 2655   class class class wbr 3893   _E cep 4167   Se wse 4209

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 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rab 2397  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-eprel 4169  df-se 4213

This theorem is referenced by: (None)