Theorem intmin

Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Assertion	intmin	|- ( A e. B -> |^| { x e. B | A C_ x } = A )

Proof

Step	Hyp	Ref	Expression
1		vex	|- y e. _V
2	1	elintrab	|- ( y e. |^| { x e. B | A C_ x } <-> A. x e. B ( A C_ x -> y e. x ) )
3		ssid	|- A C_ A
4		sseq2	|- ( x = A -> ( A C_ x <-> A C_ A ) )
5		eleq2	|- ( x = A -> ( y e. x <-> y e. A ) )
6	4 5	imbi12d	|- ( x = A -> ( ( A C_ x -> y e. x ) <-> ( A C_ A -> y e. A ) ) )
7	6	rspcv	|- ( A e. B -> ( A. x e. B ( A C_ x -> y e. x ) -> ( A C_ A -> y e. A ) ) )
8	3 7	mpii	|- ( A e. B -> ( A. x e. B ( A C_ x -> y e. x ) -> y e. A ) )
9	2 8	biimtrid	|- ( A e. B -> ( y e. |^| { x e. B | A C_ x } -> y e. A ) )
10	9	ssrdv	|- ( A e. B -> |^| { x e. B | A C_ x } C_ A )
11		ssintub	|- A C_ |^| { x e. B | A C_ x }
12	11	a1i	|- ( A e. B -> A C_ |^| { x e. B | A C_ x } )
13	10 12	eqssd	|- ( A e. B -> |^| { x e. B | A C_ x } = A )