Theorem pleval2i

Description: One direction of pleval2 . (Contributed 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	pleval2i	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(X\underset{\dot{}}{<}Y\vee X=Y\right)\right)$$

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		elfvdm	$$⊢X\in {\mathrm{Base}}_K\to K\in \mathrm{dom}\mathrm{Base}$$
5	4 1	eleq2s	$$⊢X\in B\to K\in \mathrm{dom}\mathrm{Base}$$
6	5	adantr	$$⊢\left(X\in B\wedge Y\in B\right)\to K\in \mathrm{dom}\mathrm{Base}$$
7	2 3	pltval	$$⊢\left(K\in \mathrm{dom}\mathrm{Base}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{<}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
8	7	3expb	$$⊢\left(K\in \mathrm{dom}\mathrm{Base}\wedge \left(X\in B\wedge Y\in B\right)\right)\to \left(X\underset{\dot{}}{<}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
9	6 8	mpancom	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{<}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
10	9	biimpar	$$⊢\left(\left(X\in B\wedge Y\in B\right)\wedge \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)\to X\underset{\dot{}}{<}Y$$
11	10	expr	$$⊢\left(\left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(X\ne Y\to X\underset{\dot{}}{<}Y\right)$$
12	11	necon1bd	$$⊢\left(\left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(\neg X\underset{\dot{}}{<}Y\to X=Y\right)$$
13	12	orrd	$$⊢\left(\left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(X\underset{\dot{}}{<}Y\vee X=Y\right)$$
14	13	ex	$$⊢\left(X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(X\underset{\dot{}}{<}Y\vee X=Y\right)\right)$$