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Metamath Proof Explorer

Theorem chjcl

Description: Closure of join in CH . (Contributed by NM, 2-Nov-1999) (New usage is discouraged.)

Ref Expression
Assertion chjcl ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )

Proof

Step Hyp Ref Expression
1 chsh ⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2 chsh ⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3 shjcl ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )