Theorem cvrnbtwn

Description: There is no element between the two arguments of the covers relation. ( cvnbtwn analog.) (Contributed by NM, 18-Oct-2011)

	Ref	Expression
Hypotheses	cvrfval.b	$$⊢B={\mathrm{Base}}_K$$
	cvrfval.s	$$⊢\underset{\dot{}}{<}={<}_K$$
	cvrfval.c	$$⊢C={⋖}_K$$
Assertion	cvrnbtwn	$$⊢\left(K\in A\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\wedge XCY\right)\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		cvrfval.b	$$⊢B={\mathrm{Base}}_K$$
2		cvrfval.s	$$⊢\underset{\dot{}}{<}={<}_K$$
3		cvrfval.c	$$⊢C={⋖}_K$$
4	1 2 3	cvrval	$$⊢\left(K\in A\wedge X\in B\wedge Y\in B\right)\to \left(XCY\leftrightarrow \left(X\underset{\dot{}}{<}Y\wedge \neg \exists z\in B\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\right)\right)$$
5	4	3adant3r3	$$⊢\left(K\in A\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(XCY\leftrightarrow \left(X\underset{\dot{}}{<}Y\wedge \neg \exists z\in B\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\right)\right)$$
6		ralnex	$$⊢\forall z\in B\neg \left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\leftrightarrow \neg \exists z\in B\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)$$
7		breq2	$$⊢z=Z\to \left(X\underset{\dot{}}{<}z\leftrightarrow X\underset{\dot{}}{<}Z\right)$$
8		breq1	$$⊢z=Z\to \left(z\underset{\dot{}}{<}Y\leftrightarrow Z\underset{\dot{}}{<}Y\right)$$
9	7 8	anbi12d	$$⊢z=Z\to \left(\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\leftrightarrow \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
10	9	notbid	$$⊢z=Z\to \left(\neg \left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\leftrightarrow \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
11	10	rspcv	$$⊢Z\in B\to \left(\forall z\in B\neg \left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
12	6 11	syl5bir	$$⊢Z\in B\to \left(\neg \exists z\in B\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
13	12	adantld	$$⊢Z\in B\to \left(\left(X\underset{\dot{}}{<}Y\wedge \neg \exists z\in B\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\right)\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
14	13	3ad2ant3	$$⊢\left(X\in B\wedge Y\in B\wedge Z\in B\right)\to \left(\left(X\underset{\dot{}}{<}Y\wedge \neg \exists z\in B\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\right)\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
15	14	adantl	$$⊢\left(K\in A\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{<}Y\wedge \neg \exists z\in B\left(X\underset{\dot{}}{<}z\wedge z\underset{\dot{}}{<}Y\right)\right)\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
16	5 15	sylbid	$$⊢\left(K\in A\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(XCY\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)\right)$$
17	16	3impia	$$⊢\left(K\in A\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\wedge XCY\right)\to \neg \left(X\underset{\dot{}}{<}Z\wedge Z\underset{\dot{}}{<}Y\right)$$