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Metamath Proof Explorer

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 ⊆ C_ℋ → ( ∨_ℋ ‘ 𝐴 ) ∈ C_ℋ )

Proof

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