ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqvisset Unicode version

Theorem eqvisset 2722
Description: A class equal to a variable is a set. Note the absence of disjoint variable condition, contrary to isset 2718 and issetri 2721. (Contributed by BJ, 27-Apr-2019.)
Assertion
Ref Expression
eqvisset  |-  ( x  =  A  ->  A  e.  _V )

Proof of Theorem eqvisset
StepHypRef Expression
1 vex 2715 . 2  |-  x  e. 
_V
2 eleq1 2220 . 2  |-  ( x  =  A  ->  (
x  e.  _V  <->  A  e.  _V ) )
31, 2mpbii 147 1  |-  ( x  =  A  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128   _Vcvv 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714
This theorem is referenced by:  elxp5  5074  xpsnen  6766  fival  6914
  Copyright terms: Public domain W3C validator