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Theorem isset 3456
Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 3446) 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 7451. Note that a class 𝐴 which is not a set is called a proper class. In some theorems, such as uniexg 7450, in order to shorten certain proofs we use the more general antecedent 𝐴𝑉 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 2873 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, 26-May-1993.)

Assertion
Ref Expression
isset (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem isset
StepHypRef Expression
1 dfclel 2874 . 2 (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ V))
2 vex 3447 . . . 4 𝑥 ∈ V
32biantru 533 . . 3 (𝑥 = 𝐴 ↔ (𝑥 = 𝐴𝑥 ∈ V))
43exbii 1849 . 2 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴𝑥 ∈ V))
51, 4bitr4i 281 1 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446
This theorem is referenced by:  issetf  3457  issetiOLD  3459  issetri  3460  elex  3462  elissetOLD  3465  eueq  3650  ru  3722  sbc5  3751  snprc  4616  vprc  5186  eusvnfb  5262  reusv2lem3  5269  iotaex  6308  funimaexg  6414  fvmptd3f  6764  fvmptdv2  6767  ovmpodf  7289  rankf  9211  fnpr2ob  16827  isssc  17086  snelsingles  33497  bj-snglex  34410  bj-nul  34474  dissneqlem  34758  snen1g  40229  rr-spce  40907  iotaexeu  41119  elnev  41139  ax6e2nd  41261  ax6e2ndVD  41611  ax6e2ndALT  41633  upbdrech  41934  itgsubsticclem  42614
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