Theorem trclsslem

Description: The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020)

	Ref	Expression
Assertion	trclsslem	|- ( R C_ S -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ |^| { r | ( S C_ r /\ ( r o. r ) C_ r ) } )

Proof

Step	Hyp	Ref	Expression
1		clsslem	|- ( R C_ S -> |^| { r | ( R C_ r /\ ( r o. r ) C_ r ) } C_ |^| { r | ( S C_ r /\ ( r o. r ) C_ r ) } )