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Metamath Proof Explorer

Theorem chlej1i

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 chlej1i ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )

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 shlej1i ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )