Theorem chcv2

Description: The Hilbert lattice has the covering property. (Contributed by NM, 11-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chcv2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		atelch	⊢ ( 𝐵 ∈ HAtoms → 𝐵 ∈ Cℋ )
2		chnle	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ) )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ) )
4		chcv1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )
5	3 4	bitr3d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )