Theorem onelini

Description: An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994)

	Ref	Expression
Hypothesis	on.1	$$⊢A\in \mathrm{On}$$
Assertion	onelini	$$⊢B\in A\to B=B\cap A$$

Proof

Step	Hyp	Ref	Expression
1		on.1	$$⊢A\in \mathrm{On}$$
2	1	onelssi	$$⊢B\in A\to B\subseteq A$$
3		dfss	$$⊢B\subseteq A\leftrightarrow B=B\cap A$$
4	2 3	sylib	$$⊢B\in A\to B=B\cap A$$