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Theorem isset 2729
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2727) 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 4404. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4405, 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 2249 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 2249 . 2  |-  ( A  e.  _V  <->  E. x
( x  =  A  /\  x  e.  _V ) )
2 vex 2728 . . . 4  |-  x  e. 
_V
32biantru 493 . . 3  |-  ( x  =  A  <->  ( x  =  A  /\  x  e.  _V ) )
43exbii 1580 . 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 1537    = wceq 1619    e. wcel 1621   _Vcvv 2725
This theorem is referenced by:  issetf  2730  isseti  2731  issetri  2732  elex  2733  elisset  2735  ceqex  2833  eueq  2872  moeq  2876  ru  2918  sbc5  2942  snprc  3596  vprc  4046  vnex  4048  eusvnfb  4418  reusv2lem3  4425  funimaexg  5183  fvmptdf  5460  fvmptdv2  5462  ovmpt2df  5828  iotaex  6157  rankf  7347  isssc  13503  snelsingles  23566  ceqsex3OLD  25823  iotaexeu  26716  elnev  26736  a9e2nd  26940  a9e2ndVD  27249  a9e2ndALT  27272
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-17 1628  ax-12o 1664  ax-9 1684  ax-4 1692  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-v 2727
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