Metamath Proof Explorer


Theorem chsscon1i

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

Ref Expression
Hypotheses ch0le.1 𝐴C
chjcl.2 𝐵C
Assertion chsscon1i ( ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 )

Proof

Step Hyp Ref Expression
1 ch0le.1 𝐴C
2 chjcl.2 𝐵C
3 1 choccli ( ⊥ ‘ 𝐴 ) ∈ C
4 3 2 chsscon3i ( ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) )
5 1 pjococi ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴
6 5 sseq2i ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ↔ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 )
7 4 6 bitri ( ( ⊥ ‘ 𝐴 ) ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ 𝐴 )