Theorem isset 2639

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 2635) 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 4288. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 4290, 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 2091 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 2091	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 2636	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 297	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1548	. 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 1296  ∃wex 1433   ∈ wcel 1445  Vcvv 2633

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-v 2635

This theorem is referenced by:  issetf  2640  isseti  2641  issetri  2642  elex  2644  elisset  2647  vtoclg1f  2692  ceqex  2758  eueq  2800  moeq  2804  mosubt  2806  ru  2853  sbc5  2877  snprc  3527  vprc  3992  opelopabsb  4111  eusvnfb  4304  euiotaex  5030  fvmptdf  5426  fvmptdv2  5428  fmptco  5503  brabvv  5733  ovmpt2df  5814  ovi3  5819  tfrlemibxssdm  6130  tfr1onlembxssdm  6146  tfrcllembxssdm  6159  ecexr  6337  snexxph  6739  bj-vprc  12511  bj-vnex  12513  bj-2inf  12557