Theorem ssdmd2

Description: Ordering implies the dual modular pair property. Remark in MaedaMaeda p. 1. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	ssdmd2	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		chsscon3	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) )
2		choccl	⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
3		choccl	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
4		ssmd1	⊢ ( ( ( ⊥ ‘ 𝐵 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) → ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) )
5	4	3expia	⊢ ( ( ( ⊥ ‘ 𝐵 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐴 ) ∈ Cℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) → ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) ) )
6	2 3 5	syl2anr	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) → ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) ) )
7	1 6	sylbid	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 → ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) ) )
8	7	3impia	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → ( ⊥ ‘ 𝐵 ) 𝑀ℋ ( ⊥ ‘ 𝐴 ) )