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Metamath Proof Explorer

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 )