Theorem epse 4377

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 4327	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 132	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2311	. . . . 5 |- x = { y | y _E x }
4		vex 2766	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2270	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3271	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4171	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2552	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4368	. 2 |- ( _E Se A <-> A. x e. A { y e. A | y _E x } e. _V )
10	8, 9	mpbir 146	1 |- _E Se A

Syntax hints:   e. wcel 2167   {cab 2182   A.wral 2475   {crab 2479   _Vcvv 2763   class class class wbr 4033   _E cep 4322   Se wse 4364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-eprel 4324  df-se 4368

This theorem is referenced by: (None)