Metamath Proof Explorer


Theorem mrcssvd

Description: The Moore closure of a set is a subset of the base. Deduction form of mrcssv . (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypotheses mrcssd.1
|- ( ph -> A e. ( Moore ` X ) )
mrcssd.2
|- N = ( mrCls ` A )
Assertion mrcssvd
|- ( ph -> ( N ` B ) C_ X )

Proof

Step Hyp Ref Expression
1 mrcssd.1
 |-  ( ph -> A e. ( Moore ` X ) )
2 mrcssd.2
 |-  N = ( mrCls ` A )
3 2 mrcssv
 |-  ( A e. ( Moore ` X ) -> ( N ` B ) C_ X )
4 1 3 syl
 |-  ( ph -> ( N ` B ) C_ X )