Theorem chsupcl

Description: Closure of supremum of subset of CH . Definition of supremum in Proposition 1 of Kalmbach p. 65. Shows that CH is a complete lattice. Also part of Definition 3.4-1 in MegPav2000 p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	chsupcl	⊢ ( 𝐴 ⊆ Cℋ → ( ∨ℋ ‘ 𝐴 ) ∈ Cℋ )

Proof

Step	Hyp	Ref	Expression
1		chsspwh	⊢ Cℋ ⊆ 𝒫 ℋ
2		sstr2	⊢ ( 𝐴 ⊆ Cℋ → ( Cℋ ⊆ 𝒫 ℋ → 𝐴 ⊆ 𝒫 ℋ ) )
3	1 2	mpi	⊢ ( 𝐴 ⊆ Cℋ → 𝐴 ⊆ 𝒫 ℋ )
4		hsupcl	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ( ∨ℋ ‘ 𝐴 ) ∈ Cℋ )
5	3 4	syl	⊢ ( 𝐴 ⊆ Cℋ → ( ∨ℋ ‘ 𝐴 ) ∈ Cℋ )