Description: Moore closure preserves subset ordering. Deduction form of mrcss . (Contributed by David Moews, 1-May-2017)
Ref | Expression | ||
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Hypotheses | mrcssd.1 | |- ( ph -> A e. ( Moore ` X ) ) |
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mrcssd.2 | |- N = ( mrCls ` A ) |
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mrcssd.3 | |- ( ph -> U C_ V ) |
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mrcssd.4 | |- ( ph -> V C_ X ) |
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Assertion | mrcssd | |- ( ph -> ( N ` U ) C_ ( N ` V ) ) |
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 ) |
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4 | mrcssd.4 | |- ( ph -> V C_ X ) |
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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 ) ) |