Theorem shjcl

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

	Ref	Expression
Assertion	shjcl	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) ∈ Cℋ )

Proof

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