Theorem slotsdifplendx

Description: The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfun . (Contributed by AV, 11-Nov-2024)

	Ref	Expression
Assertion	slotsdifplendx	$$⊢\left({\mathrm{*}}_{\mathrm{ndx}}\ne {\le }_{\mathrm{ndx}}\wedge \mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\le }_{\mathrm{ndx}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		4re	$$⊢4\in ℝ$$
2		4lt10	$$⊢4<10$$
3	1 2	ltneii	$$⊢4\ne 10$$
4		starvndx	$$⊢{\mathrm{*}}_{\mathrm{ndx}}=4$$
5		plendx	$$⊢{\le }_{\mathrm{ndx}}=10$$
6	4 5	neeq12i	$$⊢{\mathrm{*}}_{\mathrm{ndx}}\ne {\le }_{\mathrm{ndx}}\leftrightarrow 4\ne 10$$
7	3 6	mpbir	$$⊢{\mathrm{*}}_{\mathrm{ndx}}\ne {\le }_{\mathrm{ndx}}$$
8		9re	$$⊢9\in ℝ$$
9		9lt10	$$⊢9<10$$
10	8 9	ltneii	$$⊢9\ne 10$$
11		tsetndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)=9$$
12	11 5	neeq12i	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\le }_{\mathrm{ndx}}\leftrightarrow 9\ne 10$$
13	10 12	mpbir	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\le }_{\mathrm{ndx}}$$
14	7 13	pm3.2i	$$⊢\left({\mathrm{*}}_{\mathrm{ndx}}\ne {\le }_{\mathrm{ndx}}\wedge \mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\le }_{\mathrm{ndx}}\right)$$