Theorem epse 4344

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 4294	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 132	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2292	. . . . 5 |- x = { y | y _E x }
4		vex 2742	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2251	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3245	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4143	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2532	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4335	. 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 2148   {cab 2163   A.wral 2455   {crab 2459   _Vcvv 2739   class class class wbr 4005   _E cep 4289   Se wse 4331

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-eprel 4291  df-se 4335

This theorem is referenced by: (None)