Theorem chsupss

Description: Subset relation for supremum of subset of CH . (Contributed by NM, 13-Aug-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	chsupss	⊢ ( ( 𝐴 ⊆ Cℋ ∧ 𝐵 ⊆ Cℋ ) → ( 𝐴 ⊆ 𝐵 → ( ∨ℋ ‘ 𝐴 ) ⊆ ( ∨ℋ ‘ 𝐵 ) ) )

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		sstr2	⊢ ( 𝐵 ⊆ Cℋ → ( Cℋ ⊆ 𝒫 ℋ → 𝐵 ⊆ 𝒫 ℋ ) )
5	1 4	mpi	⊢ ( 𝐵 ⊆ Cℋ → 𝐵 ⊆ 𝒫 ℋ )
6		hsupss	⊢ ( ( 𝐴 ⊆ 𝒫 ℋ ∧ 𝐵 ⊆ 𝒫 ℋ ) → ( 𝐴 ⊆ 𝐵 → ( ∨ℋ ‘ 𝐴 ) ⊆ ( ∨ℋ ‘ 𝐵 ) ) )
7	3 5 6	syl2an	⊢ ( ( 𝐴 ⊆ Cℋ ∧ 𝐵 ⊆ Cℋ ) → ( 𝐴 ⊆ 𝐵 → ( ∨ℋ ‘ 𝐴 ) ⊆ ( ∨ℋ ‘ 𝐵 ) ) )