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Theorem isset 2539
 Description: Two ways to say "A is a set": A class A is a member of the universal class V (see df-v 2537) 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 ∈ V " to mean "A is a set" very frequently, for example in uniex . Note the when A is not a set, it is called a proper class. In some theorems, such as uniexg , in order to shorten certain proofs we use the more general antecedent A ∈ 𝑉 instead of A ∈ 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 2018 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 V ↔ x x = A)
Distinct variable group:   x,A

Proof of Theorem isset
StepHypRef Expression
1 df-clel 2018 . 2 (A V ↔ x(x = A x V))
2 vex 2538 . . . 4 x V
32biantru 286 . . 3 (x = A ↔ (x = A x V))
43exbii 1478 . 2 (x x = Ax(x = A x V))
51, 4bitr4i 176 1 (A V ↔ x x = A)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537 This theorem is referenced by:  issetf  2540  isseti  2541  issetri  2542  elex  2543  elisset  2545  ceqex  2648  eueq  2689  moeq  2693  mosubt  2695  ru  2740  sbc5  2764  snprc  3409  vprc  3862  vnex  3864  opelopabsb  3971  eusvnfb  4136  dtruex  4221  euiotaex  4810  fvmptdf  5183  fvmptdv2  5185  fmptco  5255  brabvv  5474  ovmpt2df  5555  ovi3  5560  tfrlemibxssdm  5862  ecexr  6022  bj-vprc  7119  bj-vnex  7121  bj-2inf  7160
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