Theorem shjcli

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

	Ref	Expression
Hypotheses	shincl.1	⊢ 𝐴 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
Assertion	shjcli	⊢ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ

Proof

Step	Hyp	Ref	Expression
1		shincl.1	⊢ 𝐴 ∈ Sℋ
2		shincl.2	⊢ 𝐵 ∈ Sℋ
3		shjcl	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )
4	1 2 3	mp2an	⊢ ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ