Theorem isset 2695

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 2691) 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 4367. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 4369, 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 2136 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

Step	Hyp	Ref	Expression
1		df-clel 2136	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 2692	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 300	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1585	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
5	1, 4	bitr4i 186	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691

This theorem is referenced by:  issetf  2696  isseti  2697  issetri  2698  elex  2700  elisset  2703  vtoclg1f  2748  ceqex  2816  eueq  2859  moeq  2863  mosubt  2865  ru  2912  sbc5  2936  snprc  3596  vprc  4068  opelopabsb  4190  eusvnfb  4383  euiotaex  5112  fvmptdf  5516  fvmptdv2  5518  fmptco  5594  brabvv  5825  ovmpodf  5910  ovi3  5915  tfrlemibxssdm  6232  tfr1onlembxssdm  6248  tfrcllembxssdm  6261  ecexr  6442  snexxph  6846  bj-vprc  13265  bj-vnex  13267  bj-2inf  13307