Theorem cdlemesner

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012)

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

Proof

Step	Hyp	Ref	Expression
1		cdlemesner.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdlemesner.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdlemesner.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		cdlemesner.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
5		nbrne2	$$⊢\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to R\ne S$$
6	5	3ad2ant3	$$⊢\left(K\in \mathrm{HL}\wedge \left(R\in A\wedge S\in A\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\ne S$$
7	6	necomd	$$⊢\left(K\in \mathrm{HL}\wedge \left(R\in A\wedge S\in A\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\ne R$$