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Theorem isset 2695
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2691) if and only if the class  A exists (i.e. there exists some set  x equal to class 
A). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device " A  e.  _V " to mean " A is a set" very frequently, for example in uniex 4365. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4367, in order to shorten certain proofs we use the more general antecedent  A  e.  V instead of  A  e.  _V to mean " A is a set."

Note that a constant is implicitly considered distinct from all variables. This is why  _V is not included in the distinct variable list, even though df-clel 2136 requires that the expression substituted for  B not contain  x. (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.)

Assertion
Ref Expression
isset  |-  ( A  e.  _V  <->  E. x  x  =  A )
Distinct variable group:    x, A

Proof of Theorem isset
StepHypRef Expression
1 df-clel 2136 . 2  |-  ( A  e.  _V  <->  E. x
( x  =  A  /\  x  e.  _V ) )
2 vex 2692 . . . 4  |-  x  e. 
_V
32biantru 300 . . 3  |-  ( x  =  A  <->  ( x  =  A  /\  x  e.  _V ) )
43exbii 1585 . 2  |-  ( E. x  x  =  A  <->  E. x ( x  =  A  /\  x  e. 
_V ) )
51, 4bitr4i 186 1  |-  ( A  e.  _V  <->  E. x  x  =  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1332   E.wex 1469    e. wcel 1481   _Vcvv 2689
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691
This theorem is referenced by:  issetf  2696  isseti  2697  issetri  2698  elex  2700  elisset  2703  vtoclg1f  2748  ceqex  2815  eueq  2858  moeq  2862  mosubt  2864  ru  2911  sbc5  2935  snprc  3594  vprc  4066  opelopabsb  4188  eusvnfb  4381  euiotaex  5110  fvmptdf  5514  fvmptdv2  5516  fmptco  5592  brabvv  5823  ovmpodf  5908  ovi3  5913  tfrlemibxssdm  6230  tfr1onlembxssdm  6246  tfrcllembxssdm  6259  ecexr  6440  snexxph  6844  bj-vprc  13258  bj-vnex  13260  bj-2inf  13300
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