Theorem sswf

Description: A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	sswf	|- ( ( A e. U. ( R1 " On ) /\ B C_ A ) -> B e. U. ( R1 " On ) )

Proof

Step	Hyp	Ref	Expression
1		rankidb	|- ( A e. U. ( R1 " On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
2		r1sscl	|- ( ( A e. ( R1 ` suc ( rank ` A ) ) /\ B C_ A ) -> B e. ( R1 ` suc ( rank ` A ) ) )
3	1 2	sylan	|- ( ( A e. U. ( R1 " On ) /\ B C_ A ) -> B e. ( R1 ` suc ( rank ` A ) ) )
4		r1elwf	|- ( B e. ( R1 ` suc ( rank ` A ) ) -> B e. U. ( R1 " On ) )
5	3 4	syl	|- ( ( A e. U. ( R1 " On ) /\ B C_ A ) -> B e. U. ( R1 " On ) )