# Metamath Proof Explorer

## Theorem isset

Description: Two ways to say " A is a set": A class A is a member of the universal class _V (see df-v ) 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 e.V " to mean " A is a set" very frequently, for example in uniex . Note that a class A which is not a set is called a proper class_. In some theorems, such as uniexg , in order to shorten certain proofs we use the more general antecedent A e. V instead of A e.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 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)

Ref Expression
Assertion isset ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )

### Proof

Step Hyp Ref Expression
1 dfclel ( 𝐴 ∈ V ↔ ∃ 𝑥 ( 𝑥 = 𝐴𝑥 ∈ V ) )
2 vex 𝑥 ∈ V
3 2 biantru ( 𝑥 = 𝐴 ↔ ( 𝑥 = 𝐴𝑥 ∈ V ) )
4 3 exbii ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑥 ( 𝑥 = 𝐴𝑥 ∈ V ) )
5 1 4 bitr4i ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )