Theorem epse 5097

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 5032	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 214	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2738	. . . . 5 |- x = { y | y _E x }
4		vex 3203	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2698	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3690	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 4803	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2924	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 5074	. 2 |- ( _E Se A <-> A. x e. A { y e. A | y _E x } e. _V )
10	8, 9	mpbir 221	1 |- _E Se A

Syntax hints:   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   _E cep 5028   Se wse 5071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-se 5074

This theorem is referenced by:  omsinds  7084  tfr1ALT  7496  tfr2ALT  7497  tfr3ALT  7498  oieu  8444  oismo  8445  oiid  8446  cantnfp1lem3  8577  r0weon  8835  hsmexlem1  9248