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Theorem isset 2731
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2729) 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 4407. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4408, 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 2730 . . . 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 2727
This theorem is referenced by:  issetf  2732  isseti  2733  issetri  2734  elex  2735  elisset  2737  ceqex  2835  eueq  2874  moeq  2878  ru  2920  sbc5  2945  snprc  3599  vprc  4049  vnex  4051  eusvnfb  4421  reusv2lem3  4428  funimaexg  5186  fvmptdf  5463  fvmptdv2  5465  ovmpt2df  5831  iotaex  6160  rankf  7350  isssc  13541  snelsingles  23635  ceqsex3OLD  25892  iotaexeu  26785  elnev  26805  a9e2nd  27017  a9e2ndVD  27374  a9e2ndALT  27397
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 2729
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