Theorem llnneat

Description: A lattice line is not an atom. (Contributed by NM, 19-Jun-2012)

	Ref	Expression
Hypotheses	llnneat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	llnneat.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
Assertion	llnneat	$$⊢\left(K\in \mathrm{HL}\wedge X\in N\right)\to \neg X\in A$$

Proof

Step	Hyp	Ref	Expression
1		llnneat.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		llnneat.n	$$⊢N=\mathrm{LLines}\left(K\right)$$
3		hllat	$$⊢K\in \mathrm{HL}\to K\in \mathrm{Lat}$$
4		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
5	4 2	llnbase	$$⊢X\in N\to X\in {\mathrm{Base}}_K$$
6		eqid	$$⊢{\le }_K={\le }_K$$
7	4 6	latref	$$⊢\left(K\in \mathrm{Lat}\wedge X\in {\mathrm{Base}}_K\right)\to X{\le }_{K}X$$
8	3 5 7	syl2an	$$⊢\left(K\in \mathrm{HL}\wedge X\in N\right)\to X{\le }_{K}X$$
9	6 1 2	llnnleat	$$⊢\left(K\in \mathrm{HL}\wedge X\in N\wedge X\in A\right)\to \neg X{\le }_{K}X$$
10	9	3expia	$$⊢\left(K\in \mathrm{HL}\wedge X\in N\right)\to \left(X\in A\to \neg X{\le }_{K}X\right)$$
11	8 10	mt2d	$$⊢\left(K\in \mathrm{HL}\wedge X\in N\right)\to \neg X\in A$$