Theorem atleneN

Description: Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	atlene.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	atlene.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	atlene.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	atleneN	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne R\wedge P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to Q\ne R$$

Proof

Step	Hyp	Ref	Expression
1		atlene.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		atlene.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		atlene.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		eqid	$$⊢{⋖}_K={⋖}_K$$
5	1 2 4 3	atcvrj1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne R\wedge P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to P{⋖}_K\left(Q\underset{\dot{}}{\vee }R\right)$$
6	2 4 3	atcvrneN	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge P{⋖}_K\left(Q\underset{\dot{}}{\vee }R\right)\right)\to Q\ne R$$
7	5 6	syld3an3	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(P\ne R\wedge P\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)\to Q\ne R$$