Theorem chsscon3i

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	chsscon3i	⊢ ( 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		occon	⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ⊆ 𝐵 → ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) )
6	3 4 5	mp2an	⊢ ( 𝐴 ⊆ 𝐵 → ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) )
7	2	choccli	⊢ ( ⊥ ‘ 𝐵 ) ∈ Cℋ
8	7	chssii	⊢ ( ⊥ ‘ 𝐵 ) ⊆ ℋ
9	1	choccli	⊢ ( ⊥ ‘ 𝐴 ) ∈ Cℋ
10	9	chssii	⊢ ( ⊥ ‘ 𝐴 ) ⊆ ℋ
11		occon	⊢ ( ( ( ⊥ ‘ 𝐵 ) ⊆ ℋ ∧ ( ⊥ ‘ 𝐴 ) ⊆ ℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) ) )
12	8 10 11	mp2an	⊢ ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) )
13	1	pjococi	⊢ ( ⊥ ‘ ( ⊥ ‘ 𝐴 ) ) = 𝐴
14	2	pjococi	⊢ ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝐵
15	12 13 14	3sstr3g	⊢ ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) → 𝐴 ⊆ 𝐵 )
16	6 15	impbii	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) )