Theorem chpsscon3

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

	Ref	Expression
Assertion	chpsscon3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊊ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊊ ( ⊥ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		chsscon3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) )
2		chsscon3	⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( 𝐵 ⊆ 𝐴 ↔ ( ⊥ ‘ 𝐴 ) ⊆ ( ⊥ ‘ 𝐵 ) ) )
3	2	ancoms	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐵 ⊆ 𝐴 ↔ ( ⊥ ‘ 𝐴 ) ⊆ ( ⊥ ‘ 𝐵 ) ) )
4	3	notbid	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ ¬ ( ⊥ ‘ 𝐴 ) ⊆ ( ⊥ ‘ 𝐵 ) ) )
5	1 4	anbi12d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴 ) ↔ ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ∧ ¬ ( ⊥ ‘ 𝐴 ) ⊆ ( ⊥ ‘ 𝐵 ) ) ) )
6		dfpss3	⊢ ( 𝐴 ⊊ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴 ) )
7		dfpss3	⊢ ( ( ⊥ ‘ 𝐵 ) ⊊ ( ⊥ ‘ 𝐴 ) ↔ ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ∧ ¬ ( ⊥ ‘ 𝐴 ) ⊆ ( ⊥ ‘ 𝐵 ) ) )
8	5 6 7	3bitr4g	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊊ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊊ ( ⊥ ‘ 𝐴 ) ) )