Theorem chlej2i

Description: Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	⊢ 𝐴 ∈ Cℋ
	chjcl.2	⊢ 𝐵 ∈ Cℋ
	chlub.1	⊢ 𝐶 ∈ Cℋ
Assertion	chlej2i	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3		chlub.1	⊢ 𝐶 ∈ Cℋ
4	1	chshii	⊢ 𝐴 ∈ Sℋ
5	2	chshii	⊢ 𝐵 ∈ Sℋ
6	3	chshii	⊢ 𝐶 ∈ Sℋ
7	4 5 6	shlej2i	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∨ℋ 𝐴 ) ⊆ ( 𝐶 ∨ℋ 𝐵 ) )