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Metamath Proof Explorer

Theorem shjcli

Description: Closure of CH join. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses shincl.1 ⊢ A ∈ S ℋ
shincl.2 ⊢ B ∈ S ℋ
Assertion shjcli ⊢ A ∨ ℋ B ∈ C ℋ

Proof

Step Hyp Ref Expression
1 shincl.1 ⊢ A ∈ S ℋ
2 shincl.2 ⊢ B ∈ S ℋ
3 shjcl ⊢ A ∈ S ℋ ∧ B ∈ S ℋ → A ∨ ℋ B ∈ C ℋ
4 1 2 3 mp2an ⊢ A ∨ ℋ B ∈ C ℋ