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

Theorem issetri 3064
Description: A way to say " A is a set" (inference rule). (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
issetri.1  |-  E. x  x  =  A
Assertion
Ref Expression
issetri  |-  A  e. 
_V
Distinct variable group:    x, A

Proof of Theorem issetri
StepHypRef Expression
1 issetri.1 . 2  |-  E. x  x  =  A
2 isset 3061 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
31, 2mpbir 214 1  |-  A  e. 
_V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455   E.wex 1674    e. wcel 1898   _Vcvv 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-12 1944  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059
This theorem is referenced by:  zfrep4  4537  0ex  4549  inex1  4558  pwex  4600  zfpair2  4654  uniex  6614  bj-snsetex  31602
  Copyright terms: Public domain W3C validator