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Metamath Proof Explorer

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 ) ) )