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Metamath Proof Explorer

Theorem dsndxnbasendx

Description: The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024) (Proof shortened by AV, 28-Oct-2024)

		Ref	Expression
	Assertion	dsndxnbasendx	$⊢ dist (ndx) \neq {Base}_{ndx}$

Proof

Step	Hyp	Ref	Expression
1		basendxnn	$⊢ {Base}_{ndx} \in ℕ$
2	1	nnrei	$⊢ {Base}_{ndx} \in ℝ$
3		basendxltdsndx	$⊢ {Base}_{ndx} < dist (ndx)$
4	2 3	gtneii	$⊢ dist (ndx) \neq {Base}_{ndx}$