Theorem basendxnocndx

Description: The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas . (Contributed by AV, 11-Nov-2024)

	Ref	Expression
Assertion	basendxnocndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{oc}\left(\mathrm{ndx}\right)$$

Proof

Step	Hyp	Ref	Expression
1		1re	$$⊢1\in ℝ$$
2		1nn	$$⊢1\in \mathbb{N}$$
3		1nn0	$$⊢1\in {\mathbb{N}}_0$$
4		1lt10	$$⊢1<10$$
5	2 3 3 4	declti	$$⊢1<11$$
6	1 5	ltneii	$$⊢1\ne 11$$
7		basendx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}=1$$
8		ocndx	$$⊢\mathrm{oc}\left(\mathrm{ndx}\right)=11$$
9	7 8	neeq12i	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{oc}\left(\mathrm{ndx}\right)\leftrightarrow 1\ne 11$$
10	6 9	mpbir	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{oc}\left(\mathrm{ndx}\right)$$