Theorem tsetndxnstarvndx

Description: The slot for the topology is not the slot for the involution in an extensible structure. Formerly part of proof for cnfldfun . (Contributed by AV, 11-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		4re	$$⊢4\in ℝ$$
2		4lt9	$$⊢4<9$$
3	1 2	gtneii	$$⊢9\ne 4$$
4		tsetndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)=9$$
5		starvndx	$$⊢{\mathrm{*}}_{\mathrm{ndx}}=4$$
6	4 5	neeq12i	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\mathrm{*}}_{\mathrm{ndx}}\leftrightarrow 9\ne 4$$
7	3 6	mpbir	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\mathrm{*}}_{\mathrm{ndx}}$$