Theorem otpsle

Description: The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015) (Revised by AV, 9-Sep-2021)

	Ref	Expression
Hypothesis	otpsstr.w	$$⊢K=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩\right\}$$
Assertion	otpsle	$$⊢\underset{\dot{}}{\le }\in V\to \underset{\dot{}}{\le }={\le }_K$$

Proof

Step	Hyp	Ref	Expression
1		otpsstr.w	$$⊢K=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩\right\}$$
2	1	otpsstr	$$⊢K\mathrm{Struct}⟨1,10⟩$$
3		pleid	$$⊢\mathrm{le}=\mathrm{Slot}{\le }_{\mathrm{ndx}}$$
4		snsstp3	$$⊢\left\{⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩\right\}\subseteq K$$
6	2 3 5	strfv	$$⊢\underset{\dot{}}{\le }\in V\to \underset{\dot{}}{\le }={\le }_K$$