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	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$

Proof

Step	Hyp	Ref	Expression
1		dfclel	$$⊢A\in \mathrm{V}\leftrightarrow \exists x\left(x=A\wedge x\in \mathrm{V}\right)$$
2		vex	$$⊢x\in \mathrm{V}$$
3	2	biantru	$$⊢x=A\leftrightarrow \left(x=A\wedge x\in \mathrm{V}\right)$$
4	3	exbii	$$⊢\exists xx=A\leftrightarrow \exists x\left(x=A\wedge x\in \mathrm{V}\right)$$
5	1 4	bitr4i	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$