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Metamath Proof Explorer

Theorem chsscon2

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

Ref Expression
Assertion chsscon2 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 chss ⊢ ( 𝐴 ∈ Cℋ → 𝐴 ⊆ ℋ )
2 chss ⊢ ( 𝐵 ∈ Cℋ → 𝐵 ⊆ ℋ )
3 occon3 ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ) )