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)

	Ref	Expression
Assertion	dsndxnbasendx	⊢ ( dist ‘ ndx ) ≠ ( Base ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		1nn	⊢ 1 ∈ ℕ
3		2nn0	⊢ 2 ∈ ℕ0
4		1nn0	⊢ 1 ∈ ℕ0
5		1lt10	⊢ 1 < ; 1 0
6	2 3 4 5	declti	⊢ 1 < ; 1 2
7	1 6	gtneii	⊢ ; 1 2 ≠ 1
8		dsndx	⊢ ( dist ‘ ndx ) = ; 1 2
9		basendx	⊢ ( Base ‘ ndx ) = 1
10	8 9	neeq12i	⊢ ( ( dist ‘ ndx ) ≠ ( Base ‘ ndx ) ↔ ; 1 2 ≠ 1 )
11	7 10	mpbir	⊢ ( dist ‘ ndx ) ≠ ( Base ‘ ndx )