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Metamath Proof Explorer

Theorem latnlej1l

Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012)

		Ref	Expression
	Hypotheses	latlej.b	$⊢ B = {Base}_{K}$
		latlej.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		latlej.j	$⊢ \underset{˙}{\lor} = join (K)$
	Assertion	latnlej1l	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B) \land \neg X \underset{˙}{\leq} (Y \underset{˙}{\lor} Z)) \to X \neq Y$

Proof

Step	Hyp	Ref	Expression
1		latlej.b	$⊢ B = {Base}_{K}$
2		latlej.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		latlej.j	$⊢ \underset{˙}{\lor} = join (K)$
4	1 2 3	latnlej	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B) \land \neg X \underset{˙}{\leq} (Y \underset{˙}{\lor} Z)) \to (X \neq Y \land X \neq Z)$
5	4	simpld	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B) \land \neg X \underset{˙}{\leq} (Y \underset{˙}{\lor} Z)) \to X \neq Y$