Metamath Proof Explorer

Theorem mriss

Description: An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypothesis mriss.1 
|- I = ( mrInd ` A )
Assertion mriss 
|- ( ( A e. ( Moore ` X ) /\ S e. I ) -> S C_ X )

Proof

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