atnle0

Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011)

	Ref	Expression
Hypotheses	atnle0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	atnle0.z	$⊢ \underset{˙}{0} = 0. (K)$
	atnle0.a	$⊢ A = Atoms (K)$
Assertion	atnle0	$⊢ (K \in AtLat \land P \in A) \to \neg P \underset{˙}{\leq} \underset{˙}{0}$

Step	Hyp	Ref	Expression
1		atnle0.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		atnle0.z	$⊢ \underset{˙}{0} = 0. (K)$
3		atnle0.a	$⊢ A = Atoms (K)$
4		atlpos	$⊢ K \in AtLat \to K \in Poset$
5	4	adantr	$⊢ (K \in AtLat \land P \in A) \to K \in Poset$
6		eqid	$⊢ {Base}_{K} = {Base}_{K}$
7	6 2	atl0cl	$⊢ K \in AtLat \to \underset{˙}{0} \in {Base}_{K}$
8	7	adantr	$⊢ (K \in AtLat \land P \in A) \to \underset{˙}{0} \in {Base}_{K}$
9	6 3	atbase	$⊢ P \in A \to P \in {Base}_{K}$
10	9	adantl	$⊢ (K \in AtLat \land P \in A) \to P \in {Base}_{K}$
11		eqid	$⊢ ⋖_{K} = ⋖_{K}$
12	2 11 3	atcvr0	$⊢ (K \in AtLat \land P \in A) \to \underset{˙}{0} ⋖_{K} P$
13	6 1 11	cvrnle	$⊢ ((K \in Poset \land \underset{˙}{0} \in {Base}_{K} \land P \in {Base}_{K}) \land \underset{˙}{0} ⋖_{K} P) \to \neg P \underset{˙}{\leq} \underset{˙}{0}$
14	5 8 10 12 13	syl31anc	$⊢ (K \in AtLat \land P \in A) \to \neg P \underset{˙}{\leq} \underset{˙}{0}$

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