Metamath Proof Explorer


Theorem mrcssd

Description: Moore closure preserves subset ordering. Deduction form of mrcss . (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypotheses mrcssd.1
|- ( ph -> A e. ( Moore ` X ) )
mrcssd.2
|- N = ( mrCls ` A )
mrcssd.3
|- ( ph -> U C_ V )
mrcssd.4
|- ( ph -> V C_ X )
Assertion mrcssd
|- ( ph -> ( N ` U ) C_ ( N ` V ) )

Proof

Step Hyp Ref Expression
1 mrcssd.1
 |-  ( ph -> A e. ( Moore ` X ) )
2 mrcssd.2
 |-  N = ( mrCls ` A )
3 mrcssd.3
 |-  ( ph -> U C_ V )
4 mrcssd.4
 |-  ( ph -> V C_ X )
5 2 mrcss
 |-  ( ( A e. ( Moore ` X ) /\ U C_ V /\ V C_ X ) -> ( N ` U ) C_ ( N ` V ) )
6 1 3 4 5 syl3anc
 |-  ( ph -> ( N ` U ) C_ ( N ` V ) )