Theorem epse 4433

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 4383	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 132	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2344	. . . . 5 |- x = { y | y _E x }
4		vex 2802	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2303	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3312	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4222	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2585	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4424	. 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 2200   {cab 2215   A.wral 2508   {crab 2512   _Vcvv 2799   class class class wbr 4083   _E cep 4378   Se wse 4420

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-eprel 4380  df-se 4424

This theorem is referenced by: (None)