Theorem tsetndxnbasendx

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

	Ref	Expression
Assertion	tsetndxnbasendx	⊢ ( TopSet ‘ ndx ) ≠ ( Base ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		1lt9	⊢ 1 < 9
3	1 2	gtneii	⊢ 9 ≠ 1
4		tsetndx	⊢ ( TopSet ‘ ndx ) = 9
5		basendx	⊢ ( Base ‘ ndx ) = 1
6	4 5	neeq12i	⊢ ( ( TopSet ‘ ndx ) ≠ ( Base ‘ ndx ) ↔ 9 ≠ 1 )
7	3 6	mpbir	⊢ ( TopSet ‘ ndx ) ≠ ( Base ‘ ndx )