Theorem abex

Description: Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd . (Contributed by AV, 19-Apr-2025)

	Ref	Expression
Hypotheses	abex.1	$$⊢\phi \to x\in A$$
	abex.2	$$⊢A\in \mathrm{V}$$
Assertion	abex	$$⊢\left\{x|\phi \right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		abex.1	$$⊢\phi \to x\in A$$
2		abex.2	$$⊢A\in \mathrm{V}$$
3		abss	$$⊢\left\{x|\phi \right\}\subseteq A\leftrightarrow \forall x\left(\phi \to x\in A\right)$$
4	3 1	mpgbir	$$⊢\left\{x|\phi \right\}\subseteq A$$
5	2 4	ssexi	$$⊢\left\{x|\phi \right\}\in \mathrm{V}$$