# Metamath Proof Explorer

## Theorem ssmd1

Description: Ordering implies the modular pair property. Remark in MaedaMaeda p. 1. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion ssmd1
`|- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> A MH B )`

### Proof

Step Hyp Ref Expression
1 inss1
` |-  ( ( x vH A ) i^i B ) C_ ( x vH A )`
2 dfss
` |-  ( A C_ B <-> A = ( A i^i B ) )`
3 2 biimpi
` |-  ( A C_ B -> A = ( A i^i B ) )`
4 3 oveq2d
` |-  ( A C_ B -> ( x vH A ) = ( x vH ( A i^i B ) ) )`
5 1 4 sseqtrid
` |-  ( A C_ B -> ( ( x vH A ) i^i B ) C_ ( x vH ( A i^i B ) ) )`
6 5 a1d
` |-  ( A C_ B -> ( x C_ B -> ( ( x vH A ) i^i B ) C_ ( x vH ( A i^i B ) ) ) )`
7 6 ralrimivw
` |-  ( A C_ B -> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) C_ ( x vH ( A i^i B ) ) ) )`
8 mdbr2
` |-  ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) C_ ( x vH ( A i^i B ) ) ) ) )`
9 7 8 syl5ibr
` |-  ( ( A e. CH /\ B e. CH ) -> ( A C_ B -> A MH B ) )`
10 9 3impia
` |-  ( ( A e. CH /\ B e. CH /\ A C_ B ) -> A MH B )`