Theorem basendxnocndx

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

	Ref	Expression
Assertion	basendxnocndx	⊢ ( Base ‘ ndx ) ≠ ( oc ‘ ndx )

Proof

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