Theorem pleval2

Description: "Less than or equal to" in terms of "less than". ( sspss analog.) (Contributed by NM, 17-Oct-2011) (Revised by Mario Carneiro, 8-Feb-2015)

	Ref	Expression
Hypotheses	pleval2.b	$$⊢B={\mathrm{Base}}_K$$
	pleval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	pleval2.s	$$⊢\underset{\dot{}}{<}={<}_K$$
Assertion	pleval2	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow \left(X\underset{\dot{}}{<}Y\vee X=Y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pleval2.b	$$⊢B={\mathrm{Base}}_K$$
2		pleval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		pleval2.s	$$⊢\underset{\dot{}}{<}={<}_K$$
4	1 2 3	pleval2i	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(X\underset{\dot{}}{<}Y\vee X=Y\right)\right)$$
5	4	3adant1	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(X\underset{\dot{}}{<}Y\vee X=Y\right)\right)$$
6	2 3	pltle	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{<}Y\to X\underset{\dot{}}{\le }Y\right)$$
7	1 2	posref	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\le }X$$
8	7	3adant3	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{\le }X$$
9		breq2	$$⊢X=Y\to \left(X\underset{\dot{}}{\le }X\leftrightarrow X\underset{\dot{}}{\le }Y\right)$$
10	8 9	syl5ibcom	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(X=Y\to X\underset{\dot{}}{\le }Y\right)$$
11	6 10	jaod	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{<}Y\vee X=Y\right)\to X\underset{\dot{}}{\le }Y\right)$$
12	5 11	impbid	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\leftrightarrow \left(X\underset{\dot{}}{<}Y\vee X=Y\right)\right)$$