Theorem plendxnbasendx

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

	Ref	Expression
Assertion	plendxnbasendx	⊢ ( le ‘ ndx ) ≠ ( Base ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		1lt10	⊢ 1 < ; 1 0
3	1 2	gtneii	⊢ ; 1 0 ≠ 1
4		plendx	⊢ ( le ‘ ndx ) = ; 1 0
5		basendx	⊢ ( Base ‘ ndx ) = 1
6	4 5	neeq12i	⊢ ( ( le ‘ ndx ) ≠ ( Base ‘ ndx ) ↔ ; 1 0 ≠ 1 )
7	3 6	mpbir	⊢ ( le ‘ ndx ) ≠ ( Base ‘ ndx )