Theorem chpsscon2

Description: Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chpsscon2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊊ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊊ ( ⊥ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
2		chpsscon3	⊢ ( ( 𝐴 ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ∈ Cℋ ) → ( 𝐴 ⊊ ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ⊊ ( ⊥ ‘ 𝐴 ) ) )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊊ ( ⊥ ‘ 𝐵 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ⊊ ( ⊥ ‘ 𝐴 ) ) )
4		ococ	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
5	4	adantl	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵 )
6	5	psseq1d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ⊊ ( ⊥ ‘ 𝐴 ) ↔ 𝐵 ⊊ ( ⊥ ‘ 𝐴 ) ) )
7	3 6	bitrd	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊊ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊊ ( ⊥ ‘ 𝐴 ) ) )