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Metamath Proof Explorer

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 ) ) )