Theorem epse 4320

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 4270	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 131	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2281	. . . . 5 |- x = { y | y _E x }
4		vex 2729	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2240	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3230	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4120	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2521	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4311	. 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 2136   {cab 2151   A.wral 2444   {crab 2448   _Vcvv 2726   class class class wbr 3982   _E cep 4265   Se wse 4307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-eprel 4267  df-se 4311

This theorem is referenced by: (None)