Theorem plendxnocndx

Description: The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle . (Contributed by AV, 11-Nov-2024)

	Ref	Expression
Assertion	plendxnocndx	⊢ ( le ‘ ndx ) ≠ ( oc ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		10re	⊢ ; 1 0 ∈ ℝ
2		1nn0	⊢ 1 ∈ ℕ0
3		0nn0	⊢ 0 ∈ ℕ0
4		1nn	⊢ 1 ∈ ℕ
5		0lt1	⊢ 0 < 1
6	2 3 4 5	declt	⊢ ; 1 0 < ; 1 1
7	1 6	ltneii	⊢ ; 1 0 ≠ ; 1 1
8		plendx	⊢ ( le ‘ ndx ) = ; 1 0
9		ocndx	⊢ ( oc ‘ ndx ) = ; 1 1
10	8 9	neeq12i	⊢ ( ( le ‘ ndx ) ≠ ( oc ‘ ndx ) ↔ ; 1 0 ≠ ; 1 1 )
11	7 10	mpbir	⊢ ( le ‘ ndx ) ≠ ( oc ‘ ndx )