Theorem epse 4439

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 4389	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 132	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2346	. . . . 5 |- x = { y | y _E x }
4		vex 2805	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2305	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3315	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4227	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2587	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4430	. 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 2202   {cab 2217   A.wral 2510   {crab 2514   _Vcvv 2802   class class class wbr 4088   _E cep 4384   Se wse 4426

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-eprel 4386  df-se 4430

This theorem is referenced by: (None)