# 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 ${⊢}{A}\in \mathrm{V}↔\exists {x}\phantom{\rule{.4em}{0ex}}{x}={A}$

### Proof

Step Hyp Ref Expression
1 dfclel ${⊢}{A}\in \mathrm{V}↔\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={A}\wedge {x}\in \mathrm{V}\right)$
2 vex ${⊢}{x}\in \mathrm{V}$
3 2 biantru ${⊢}{x}={A}↔\left({x}={A}\wedge {x}\in \mathrm{V}\right)$
4 3 exbii ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}{x}={A}↔\exists {x}\phantom{\rule{.4em}{0ex}}\left({x}={A}\wedge {x}\in \mathrm{V}\right)$
5 1 4 bitr4i ${⊢}{A}\in \mathrm{V}↔\exists {x}\phantom{\rule{.4em}{0ex}}{x}={A}$