Theorem dsndxntsetndx

Description: The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds . (Contributed by AV, 29-Oct-2024)

	Ref	Expression
Assertion	dsndxntsetndx	⊢ ( dist ‘ ndx ) ≠ ( TopSet ‘ ndx )

Proof

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