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Theorem isset 2794
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2792) 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 4516. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4517, 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 2281 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 2281 . 2  |-  ( A  e.  _V  <->  E. x
( x  =  A  /\  x  e.  _V ) )
2 vex 2793 . . . 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 2790
This theorem is referenced by:  issetf  2795  isseti  2796  issetri  2797  elex  2798  elisset  2800  ceqex  2900  eueq  2939  moeq  2943  ru  2992  sbc5  3017  snprc  3697  vprc  4154  vnex  4156  eusvnfb  4530  reusv2lem3  4537  funimaexg  5295  fvmptdf  5573  fvmptdv2  5575  ovmpt2df  5941  iotaex  6270  rankf  7462  isssc  13692  snelsingles  23869  ceqsex3OLD  26126  iotaexeu  27018  elnev  27038  a9e2nd  27596  a9e2ndVD  27953  a9e2ndALT  27976
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 2266
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1530  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-v 2792
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