Theorem ssab

Description: Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006)

	Ref	Expression
Assertion	ssab	$$⊢A\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		abid2	$$⊢\left\{x|x\in A\right\}=A$$
2	1	sseq1i	$$⊢\left\{x|x\in A\right\}\subseteq \left\{x|\phi \right\}\leftrightarrow A\subseteq \left\{x|\phi \right\}$$
3		ss2ab	$$⊢\left\{x|x\in A\right\}\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$
4	2 3	bitr3i	$$⊢A\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$