Theorem epse 4438

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 4388	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 132	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2345	. . . . 5 |- x = { y | y _E x }
4		vex 2804	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2304	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3314	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4226	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2586	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4429	. 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 2201   {cab 2216   A.wral 2509   {crab 2513   _Vcvv 2801   class class class wbr 4087   _E cep 4383   Se wse 4425

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rab 2518  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-br 4088  df-opab 4150  df-eprel 4385  df-se 4429

This theorem is referenced by: (None)