Theorem chsscon1

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

	Ref	Expression
Assertion	chsscon1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
2		chsscon3	⊢ ( ( ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ) )
4		ococ	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )
5	4	adantr	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴 )
6	5	sseq2d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ↔ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 ) )
7	3 6	bitrd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 ) )