Theorem chlubi

Description: Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	⊢ 𝐴 ∈ Cℋ
	chjcl.2	⊢ 𝐵 ∈ Cℋ
	chlub.1	⊢ 𝐶 ∈ Cℋ
Assertion	chlubi	⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ↔ ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3		chlub.1	⊢ 𝐶 ∈ Cℋ
4	1	chshii	⊢ 𝐴 ∈ Sℋ
5	2	chshii	⊢ 𝐵 ∈ Sℋ
6	4 5 3	shlubi	⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ↔ ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐶 )