Theorem cvpss

Description: The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	cvpss	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		cvbr	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∧ ¬ ∃ 𝑥 ∈ Cℋ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) ) ) )
2		simpl	⊢ ( ( 𝐴 ⊊ 𝐵 ∧ ¬ ∃ 𝑥 ∈ Cℋ ( 𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵 ) ) → 𝐴 ⊊ 𝐵 )
3	1 2	biimtrdi	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵 ) )