Theorem epse 4079

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 4029	. . . . . . 7 |- ( y _E x <-> y e. x )
2	1	bicomi 123	. . . . . 6 |- ( y e. x <-> y _E x )
3	2	abbi2i 2152	. . . . 5 |- x = { y | y _E x }
4		vex 2560	. . . . 5 |- x e. _V
5	3, 4	eqeltrri 2111	. . . 4 |- { y | y _E x } e. _V
6		rabssab 3027	. . . 4 |- { y e. A | y _E x } C_ { y | y _E x }
7	5, 6	ssexi 3895	. . 3 |- { y e. A | y _E x } e. _V
8	7	rgenw 2376	. 2 |- A. x e. A { y e. A | y _E x } e. _V
9		df-se 4070	. 2 |- ( _E Se A <-> A. x e. A { y e. A | y _E x } e. _V )
10	8, 9	mpbir 134	1 |- _E Se A

Syntax hints:   e. wcel 1393   {cab 2026   A.wral 2306   {crab 2310   _Vcvv 2557   class class class wbr 3764   _E cep 4024   Se wse 4066

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026  df-se 4070

This theorem is referenced by: (None)