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Theorem isset 2805
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2803) 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 4532. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4533, 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 2292 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 2292 . 2  |-  ( A  e.  _V  <->  E. x
( x  =  A  /\  x  e.  _V ) )
2 vex 2804 . . . 4  |-  x  e. 
_V
32biantru 491 . . 3  |-  ( x  =  A  <->  ( x  =  A  /\  x  e.  _V ) )
43exbii 1572 . 2  |-  ( E. x  x  =  A  <->  E. x ( x  =  A  /\  x  e. 
_V ) )
51, 4bitr4i 243 1  |-  ( A  e.  _V  <->  E. x  x  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem is referenced by:  issetf  2806  isseti  2807  issetri  2808  elex  2809  elisset  2811  ceqex  2911  eueq  2950  moeq  2954  ru  3003  sbc5  3028  snprc  3708  vprc  4168  vnex  4170  eusvnfb  4546  reusv2lem3  4553  iotaex  5252  funimaexg  5345  fvmptdf  5627  fvmptdv2  5629  ovmpt2df  5995  rankf  7482  isssc  13713  snelsingles  24531  ceqsex3OLD  26828  iotaexeu  27720  elnev  27740  a9e2nd  28622  a9e2ndVD  28999  a9e2ndALT  29022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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