Theorem plendxnbasendx

Description: The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024)

	Ref	Expression
Assertion	plendxnbasendx	$$⊢{\le }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		1re	$$⊢1\in ℝ$$
2		1lt10	$$⊢1<10$$
3	1 2	gtneii	$$⊢10\ne 1$$
4		plendx	$$⊢{\le }_{\mathrm{ndx}}=10$$
5		basendx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}=1$$
6	4 5	neeq12i	$$⊢{\le }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}\leftrightarrow 10\ne 1$$
7	3 6	mpbir	$$⊢{\le }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$