Theorem intminss

Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013)

	Ref	Expression
Hypothesis	intminss.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	intminss	$$⊢\left(A\in B\wedge \psi \right)\to \bigcap \left\{x\in B|\phi \right\}\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		intminss.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	elrab	$$⊢A\in \left\{x\in B|\phi \right\}\leftrightarrow \left(A\in B\wedge \psi \right)$$
3		intss1	$$⊢A\in \left\{x\in B|\phi \right\}\to \bigcap \left\{x\in B|\phi \right\}\subseteq A$$
4	2 3	sylbir	$$⊢\left(A\in B\wedge \psi \right)\to \bigcap \left\{x\in B|\phi \right\}\subseteq A$$