This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem occon2i

Description: Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

Ref Expression
Hypotheses occon2.1 ⊢ 𝐴 ⊆ ℋ
occon2.2 ⊢ 𝐵 ⊆ ℋ
Assertion occon2i ( 𝐴 ⊆ 𝐵 → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 occon2.1 ⊢ 𝐴 ⊆ ℋ
2 occon2.2 ⊢ 𝐵 ⊆ ℋ
3 occon2 ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ⊆ 𝐵 → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
4 1 2 3 mp2an ⊢ ( 𝐴 ⊆ 𝐵 → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) )