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Theorem isset 2793
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2791) 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 4515. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4516, 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 2280 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, 5-Aug-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 2280 . 2  |-  ( A  e.  _V  <->  E. x
( x  =  A  /\  x  e.  _V ) )
2 vex 2792 . . . 4  |-  x  e. 
_V
32biantru 493 . . 3  |-  ( x  =  A  <->  ( x  =  A  /\  x  e.  _V ) )
43exbii 1570 . 2  |-  ( E. x  x  =  A  <->  E. x ( x  =  A  /\  x  e. 
_V ) )
51, 4bitr4i 245 1  |-  ( A  e.  _V  <->  E. x  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   _Vcvv 2789
This theorem is referenced by:  issetf  2794  isseti  2795  issetri  2796  elex  2797  elisset  2799  ceqex  2899  eueq  2938  moeq  2942  ru  2991  sbc5  3016  snprc  3696  vprc  4153  vnex  4155  eusvnfb  4529  reusv2lem3  4536  funimaexg  5294  fvmptdf  5572  fvmptdv2  5574  ovmpt2df  5940  iotaex  6269  rankf  7461  isssc  13691  snelsingles  23868  ceqsex3OLD  26125  iotaexeu  27017  elnev  27037  a9e2nd  27595  a9e2ndVD  27952  a9e2ndALT  27975
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-11 1716  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-v 2791
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