Theorem isset 2806

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 2801) 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 4528. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 4530, 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 2225 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 2225	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 2802	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 302	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1651	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
5	1, 4	bitr4i 187	1 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  issetf  2807  isseti  2808  issetri  2809  elex  2811  elisset  2814  vtoclg1f  2860  ceqex  2930  eueq  2974  moeq  2978  mosubt  2980  ru  3027  sbc5  3052  snprc  3731  snmb  3788  snssb  3801  vprc  4216  opelopabsb  4348  eusvnfb  4545  elrelimasn  5094  euiotaex  5295  fvmptdf  5724  fvmptdv2  5726  fmptco  5803  brabvv  6056  ovmpodf  6142  ovi3  6148  tfrlemibxssdm  6479  tfr1onlembxssdm  6495  tfrcllembxssdm  6508  ecexr  6693  snexxph  7128  fnpr2ob  13388  bj-vprc  16314  bj-vnex  16316  bj-2inf  16356