Theorem onelss

Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	onelss	|- ( A e. On -> ( B e. A -> B C_ A ) )

Proof

Step	Hyp	Ref	Expression
1		eloni	|- ( A e. On -> Ord A )
2		ordelss	|- ( ( Ord A /\ B e. A ) -> B C_ A )
3	2	ex	|- ( Ord A -> ( B e. A -> B C_ A ) )
4	1 3	syl	|- ( A e. On -> ( B e. A -> B C_ A ) )