This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )