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Theorem isset 2767
Description: Two ways to say " A is a set": A class  A is a member of the universal class  _V (see df-v 2765) 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 4488. Note the when  A is not a set, it is called a proper class. In some theorems, such as uniexg 4489, 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 2254 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 2254 . 2  |-  ( A  e.  _V  <->  E. x
( x  =  A  /\  x  e.  _V ) )
2 vex 2766 . . . 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 2763
This theorem is referenced by:  issetf  2768  isseti  2769  issetri  2770  elex  2771  elisset  2773  ceqex  2873  eueq  2912  moeq  2916  ru  2965  sbc5  2990  snprc  3669  vprc  4126  vnex  4128  eusvnfb  4502  reusv2lem3  4509  funimaexg  5267  fvmptdf  5545  fvmptdv2  5547  ovmpt2df  5913  iotaex  6242  rankf  7434  isssc  13659  snelsingles  23836  ceqsex3OLD  26093  iotaexeu  26986  elnev  27006  a9e2nd  27377  a9e2ndVD  27734  a9e2ndALT  27757
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 2239
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-v 2765
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