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) (Proof shortened by AV, 30-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		basendxnn	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\in \mathbb{N}$$
2	1	nnrei	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\in ℝ$$
3		basendxltplendx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<{\le }_{\mathrm{ndx}}$$
4	2 3	gtneii	$$⊢{\le }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$