Theorem ismri2

Description: Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	ismri2.1	|- N = ( mrCls ` A )
	ismri2.2	|- I = ( mrInd ` A )
Assertion	ismri2	|- ( ( A e. ( Moore ` X ) /\ S C_ X ) -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ismri2.1	|- N = ( mrCls ` A )
2		ismri2.2	|- I = ( mrInd ` A )
3	1 2	ismri	|- ( A e. ( Moore ` X ) -> ( S e. I <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )
4	3	baibd	|- ( ( A e. ( Moore ` X ) /\ S C_ X ) -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )