Theorem pltval

Description: Less-than relation. ( df-pss analog.) (Contributed by NM, 12-Oct-2011)

	Ref	Expression
Hypotheses	pltval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	pltval.s	$$⊢\underset{\dot{}}{<}={<}_K$$
Assertion	pltval	$$⊢\left(K\in A\wedge X\in B\wedge Y\in C\right)\to \left(X\underset{\dot{}}{<}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pltval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		pltval.s	$$⊢\underset{\dot{}}{<}={<}_K$$
3	1 2	pltfval	$$⊢K\in A\to \underset{\dot{}}{<}=\underset{\dot{}}{\le }\setminus \mathrm{I}$$
4	3	breqd	$$⊢K\in A\to \left(X\underset{\dot{}}{<}Y\leftrightarrow X\left(\underset{\dot{}}{\le }\setminus \mathrm{I}\right)Y\right)$$
5		brdif	$$⊢X\left(\underset{\dot{}}{\le }\setminus \mathrm{I}\right)Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge \neg X\mathrm{I}Y\right)$$
6		ideqg	$$⊢Y\in C\to \left(X\mathrm{I}Y\leftrightarrow X=Y\right)$$
7	6	necon3bbid	$$⊢Y\in C\to \left(\neg X\mathrm{I}Y\leftrightarrow X\ne Y\right)$$
8	7	adantl	$$⊢\left(X\in B\wedge Y\in C\right)\to \left(\neg X\mathrm{I}Y\leftrightarrow X\ne Y\right)$$
9	8	anbi2d	$$⊢\left(X\in B\wedge Y\in C\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge \neg X\mathrm{I}Y\right)\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
10	5 9	syl5bb	$$⊢\left(X\in B\wedge Y\in C\right)\to \left(X\left(\underset{\dot{}}{\le }\setminus \mathrm{I}\right)Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
11	4 10	sylan9bb	$$⊢\left(K\in A\wedge \left(X\in B\wedge Y\in C\right)\right)\to \left(X\underset{\dot{}}{<}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
12	11	3impb	$$⊢\left(K\in A\wedge X\in B\wedge Y\in C\right)\to \left(X\underset{\dot{}}{<}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$