MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  epse Unicode version

Theorem epse 4376
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.
StepHypRef Expression
1 epel 4308 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
21bicomi 193 . . . . . 6  |-  ( y  e.  x  <->  y  _E  x )
32abbi2i 2394 . . . . 5  |-  x  =  { y  |  y  _E  x }
4 vex 2791 . . . . 5  |-  x  e. 
_V
53, 4eqeltrri 2354 . . . 4  |-  { y  |  y  _E  x }  e.  _V
6 dfrab2 3443 . . . . 5  |-  { y  e.  A  |  y  _E  x }  =  ( { y  |  y  _E  x }  i^i  A )
7 inss1 3389 . . . . 5  |-  ( { y  |  y  _E  x }  i^i  A
)  C_  { y  |  y  _E  x }
86, 7eqsstri 3208 . . . 4  |-  { y  e.  A  |  y  _E  x }  C_  { y  |  y  _E  x }
95, 8ssexi 4159 . . 3  |-  { y  e.  A  |  y  _E  x }  e.  _V
109rgenw 2610 . 2  |-  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V
11 df-se 4353 . 2  |-  (  _E Se 
A  <->  A. x  e.  A  { y  e.  A  |  y  _E  x }  e.  _V )
1210, 11mpbir 200 1  |-  _E Se  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   {cab 2269   A.wral 2543   {crab 2547   _Vcvv 2788    i^i cin 3151   class class class wbr 4023    _E cep 4303   Se wse 4350
This theorem is referenced by:  oieu  7254  oismo  7255  oiid  7256  cantnfp1lem3  7382  r0weon  7640  hsmexlem1  8052  omsinds  24219  tfr1ALT  24277  tfr2ALT  24278  tfr3ALT  24279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-se 4353
  Copyright terms: Public domain W3C validator