Theorem cvnbtwn3

Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvnbtwn3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) → 𝐶 = 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cvnbtwn	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → ¬ ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ) )
2		iman	⊢ ( ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) → 𝐴 = 𝐶 ) ↔ ¬ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ∧ ¬ 𝐴 = 𝐶 ) )
3		eqcom	⊢ ( 𝐶 = 𝐴 ↔ 𝐴 = 𝐶 )
4	3	imbi2i	⊢ ( ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) → 𝐶 = 𝐴 ) ↔ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) → 𝐴 = 𝐶 ) )
5		dfpss2	⊢ ( 𝐴 ⊊ 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶 ) )
6	5	anbi1i	⊢ ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ↔ ( ( 𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶 ) ∧ 𝐶 ⊊ 𝐵 ) )
7		an32	⊢ ( ( ( 𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶 ) ∧ 𝐶 ⊊ 𝐵 ) ↔ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ∧ ¬ 𝐴 = 𝐶 ) )
8	6 7	bitri	⊢ ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ↔ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ∧ ¬ 𝐴 = 𝐶 ) )
9	8	notbii	⊢ ( ¬ ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ↔ ¬ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ∧ ¬ 𝐴 = 𝐶 ) )
10	2 4 9	3bitr4ri	⊢ ( ¬ ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ↔ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) → 𝐶 = 𝐴 ) )
11	1 10	syl6ib	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) → 𝐶 = 𝐴 ) ) )