Theorem slotsdifplendx2

Description: The index of the slot for the "less than or equal to" ordering is not the index of other slots. Formerly part of proof for prstcleval . (Contributed by AV, 12-Nov-2024)

	Ref	Expression
Assertion	slotsdifplendx2	⊢ ( ( le ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( le ‘ ndx ) ≠ ( Hom ‘ ndx ) )

Proof

Step	Hyp	Ref	Expression
1		10re	⊢ ; 1 0 ∈ ℝ
2		1nn0	⊢ 1 ∈ ℕ0
3		0nn0	⊢ 0 ∈ ℕ0
4		5nn	⊢ 5 ∈ ℕ
5		5pos	⊢ 0 < 5
6	2 3 4 5	declt	⊢ ; 1 0 < ; 1 5
7	1 6	ltneii	⊢ ; 1 0 ≠ ; 1 5
8		plendx	⊢ ( le ‘ ndx ) = ; 1 0
9		ccondx	⊢ ( comp ‘ ndx ) = ; 1 5
10	8 9	neeq12i	⊢ ( ( le ‘ ndx ) ≠ ( comp ‘ ndx ) ↔ ; 1 0 ≠ ; 1 5 )
11	7 10	mpbir	⊢ ( le ‘ ndx ) ≠ ( comp ‘ ndx )
12		4nn	⊢ 4 ∈ ℕ
13		4pos	⊢ 0 < 4
14	2 3 12 13	declt	⊢ ; 1 0 < ; 1 4
15	1 14	ltneii	⊢ ; 1 0 ≠ ; 1 4
16		homndx	⊢ ( Hom ‘ ndx ) = ; 1 4
17	8 16	neeq12i	⊢ ( ( le ‘ ndx ) ≠ ( Hom ‘ ndx ) ↔ ; 1 0 ≠ ; 1 4 )
18	15 17	mpbir	⊢ ( le ‘ ndx ) ≠ ( Hom ‘ ndx )
19	11 18	pm3.2i	⊢ ( ( le ‘ ndx ) ≠ ( comp ‘ ndx ) ∧ ( le ‘ ndx ) ≠ ( Hom ‘ ndx ) )