Theorem chlejb2

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlejb2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∨ℋ 𝐴 ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		chlejb1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 ) )
2		chjcom	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
3	2	eqeq1d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 ↔ ( 𝐵 ∨ℋ 𝐴 ) = 𝐵 ) )
4	1 3	bitrd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∨ℋ 𝐴 ) = 𝐵 ) )