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Metamath Proof Explorer

Theorem ssdmd1

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 ssdmd1 ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 𝑀ℋ* 𝐵 )

Proof

Step Hyp Ref Expression
1 choccl ⊢ ( 𝐵 ∈ Cℋ → ( ⊥ ‘ 𝐵 ) ∈ Cℋ )
2 choccl ⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
3 ssmd2 ⊢ ( ( ( ⊥ ‘ 𝐵 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐴 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) )
4 3 3expia ⊢ ( ( ( ⊥ ‘ 𝐵 ) ∈ Cℋ ∧ ( ⊥ ‘ 𝐴 ) ∈ Cℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
5 1 2 4 syl2anr ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) → ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
6 chsscon3 ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( ⊥ ‘ 𝐵 ) ⊆ ( ⊥ ‘ 𝐴 ) ) )
7 dmdmd ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 𝑀ℋ* 𝐵 ↔ ( ⊥ ‘ 𝐴 ) 𝑀ℋ ( ⊥ ‘ 𝐵 ) ) )
8 5 6 7 3imtr4d ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 → 𝐴 𝑀ℋ* 𝐵 ) )
9 8 3impia ⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 𝑀ℋ* 𝐵 )