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 ) ) )