Theorem isset 2692

Description: Two ways to say "𝐴 is a set": A class 𝐴 is a member of the universal class V (see df-v 2688) 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 4359. Note the when 𝐴 is not a set, it is called a proper class. In some theorems, such as uniexg 4361, 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 2135 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 2135	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ V))
2		vex 2689	. . . 4 ⊢ 𝑥 ∈ V
3	2	biantru 300	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ V))
4	3	exbii 1584	. 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 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688

This theorem is referenced by:  issetf  2693  isseti  2694  issetri  2695  elex  2697  elisset  2700  vtoclg1f  2745  ceqex  2812  eueq  2855  moeq  2859  mosubt  2861  ru  2908  sbc5  2932  snprc  3588  vprc  4060  opelopabsb  4182  eusvnfb  4375  euiotaex  5104  fvmptdf  5508  fvmptdv2  5510  fmptco  5586  brabvv  5817  ovmpodf  5902  ovi3  5907  tfrlemibxssdm  6224  tfr1onlembxssdm  6240  tfrcllembxssdm  6253  ecexr  6434  snexxph  6838  bj-vprc  13094  bj-vnex  13096  bj-2inf  13136