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Metamath Proof Explorer

Theorem chsscon2i

Description: Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses ch0le.1 ⊢ 𝐴 ∈ Cℋ
chjcl.2 ⊢ 𝐵 ∈ Cℋ
Assertion chsscon2i ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ch0le.1 ⊢ 𝐴 ∈ Cℋ
2 chjcl.2 ⊢ 𝐵 ∈ Cℋ
3 1 chssii ⊢ 𝐴 ⊆ ℋ
4 2 chssii ⊢ 𝐵 ⊆ ℋ
5 occon3 ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) ) )
6 3 4 5 mp2an ⊢ ( 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ↔ 𝐵 ⊆ ( ⊥ ‘ 𝐴 ) )