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Theorem isset 2761
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2759) 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 4474. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4475, 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 2252 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 2252 . 2  |-  ( A  e.  _V  <->  E. x
( x  =  A  /\  x  e.  _V ) )
2 vex 2760 . . . 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 2757
This theorem is referenced by:  issetf  2762  isseti  2763  issetri  2764  elex  2765  elisset  2767  ceqex  2866  eueq  2905  moeq  2909  ru  2951  sbc5  2976  snprc  3655  vprc  4112  vnex  4114  eusvnfb  4488  reusv2lem3  4495  funimaexg  5253  fvmptdf  5531  fvmptdv2  5533  ovmpt2df  5899  iotaex  6228  rankf  7420  isssc  13645  snelsingles  23822  ceqsex3OLD  26079  iotaexeu  26972  elnev  26992  a9e2nd  27361  a9e2ndVD  27718  a9e2ndALT  27741
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 2237
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-v 2759
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