Theorem tcid

Description: Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013)

	Ref	Expression
Assertion	tcid	|- ( A e. V -> A C_ ( TC ` A ) )

Proof

Step	Hyp	Ref	Expression
1		ssmin	|- A C_ |^| { x | ( A C_ x /\ Tr x ) }
2		tcvalg	|- ( A e. V -> ( TC ` A ) = |^| { x | ( A C_ x /\ Tr x ) } )
3	1 2	sseqtrrid	|- ( A e. V -> A C_ ( TC ` A ) )