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 ) ≠ ( Base ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		basendxnn	⊢ ( Base ‘ ndx ) ∈ ℕ
2	1	nnrei	⊢ ( Base ‘ ndx ) ∈ ℝ
3		basendxltdsndx	⊢ ( Base ‘ ndx ) < ( dist ‘ ndx )
4	2 3	gtneii	⊢ ( dist ‘ ndx ) ≠ ( Base ‘ ndx )