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Metamath Proof Explorer

Theorem shlej1i

Description: Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999) (Revised by Mario Carneiro, 15-May-2014) (New usage is discouraged.)

Ref Expression
Hypotheses shincl.1 ⊢ 𝐴 ∈ Sℋ
shincl.2 ⊢ 𝐵 ∈ Sℋ
shless.1 ⊢ 𝐶 ∈ Sℋ
Assertion shlej1i ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 shincl.1 ⊢ 𝐴 ∈ Sℋ
2 shincl.2 ⊢ 𝐵 ∈ Sℋ
3 shless.1 ⊢ 𝐶 ∈ Sℋ
4 shlej1 ⊢ ( ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )
5 4 ex ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) ) )
6 1 2 3 5 mp3an ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐶 ) ⊆ ( 𝐵 ∨ℋ 𝐶 ) )