Theorem chsscon2

Description: Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chsscon2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		chss	⊢ ( 𝐴 ∈ Cℋ → 𝐴 ⊆ ℋ )
2		chss	⊢ ( 𝐵 ∈ Cℋ → 𝐵 ⊆ ℋ )
3		occon3	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ) )