Metamath Proof Explorer

Theorem dmdbr2

Description: Binary relation expressing the dual modular pair property. This version has a weaker constraint than dmdbr . (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion dmdbr2 
|- ( ( A e. CH /\ B e. CH ) -> ( A MH* B <-> A. x e. CH ( B C_ x -> ( x i^i ( A vH B ) ) C_ ( ( x i^i A ) vH B ) ) ) )

Proof

Step Hyp Ref Expression
1 dmdbr 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A MH* B <-> A. x e. CH ( B C_ x -> ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) ) ) )
2 chincl 
 |-  ( ( x e. CH /\ A e. CH ) -> ( x i^i A ) e. CH )
3 2 ancoms 
 |-  ( ( A e. CH /\ x e. CH ) -> ( x i^i A ) e. CH )
4 3 adantlr 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> ( x i^i A ) e. CH )
5 simplr 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> B e. CH )
6 simpr 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> x e. CH )
7 inss1 
 |-  ( x i^i A ) C_ x
8 chlub 
 |-  ( ( ( x i^i A ) e. CH /\ B e. CH /\ x e. CH ) -> ( ( ( x i^i A ) C_ x /\ B C_ x ) <-> ( ( x i^i A ) vH B ) C_ x ) )
9 8 biimpd 
 |-  ( ( ( x i^i A ) e. CH /\ B e. CH /\ x e. CH ) -> ( ( ( x i^i A ) C_ x /\ B C_ x ) -> ( ( x i^i A ) vH B ) C_ x ) )
10 7 9 mpani 
 |-  ( ( ( x i^i A ) e. CH /\ B e. CH /\ x e. CH ) -> ( B C_ x -> ( ( x i^i A ) vH B ) C_ x ) )
11 4 5 6 10 syl3anc 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> ( B C_ x -> ( ( x i^i A ) vH B ) C_ x ) )
12 simpll 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> A e. CH )
13 inss2 
 |-  ( x i^i A ) C_ A
14 chlej1 
 |-  ( ( ( ( x i^i A ) e. CH /\ A e. CH /\ B e. CH ) /\ ( x i^i A ) C_ A ) -> ( ( x i^i A ) vH B ) C_ ( A vH B ) )
15 13 14 mpan2 
 |-  ( ( ( x i^i A ) e. CH /\ A e. CH /\ B e. CH ) -> ( ( x i^i A ) vH B ) C_ ( A vH B ) )
16 4 12 5 15 syl3anc 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> ( ( x i^i A ) vH B ) C_ ( A vH B ) )
17 11 16 jctird 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> ( B C_ x -> ( ( ( x i^i A ) vH B ) C_ x /\ ( ( x i^i A ) vH B ) C_ ( A vH B ) ) ) )
18 ssin 
 |-  ( ( ( ( x i^i A ) vH B ) C_ x /\ ( ( x i^i A ) vH B ) C_ ( A vH B ) ) <-> ( ( x i^i A ) vH B ) C_ ( x i^i ( A vH B ) ) )
19 17 18 syl6ib 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> ( B C_ x -> ( ( x i^i A ) vH B ) C_ ( x i^i ( A vH B ) ) ) )
20 eqss 
 |-  ( ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) <-> ( ( ( x i^i A ) vH B ) C_ ( x i^i ( A vH B ) ) /\ ( x i^i ( A vH B ) ) C_ ( ( x i^i A ) vH B ) ) )
21 20 baib 
 |-  ( ( ( x i^i A ) vH B ) C_ ( x i^i ( A vH B ) ) -> ( ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) <-> ( x i^i ( A vH B ) ) C_ ( ( x i^i A ) vH B ) ) )
22 19 21 syl6 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> ( B C_ x -> ( ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) <-> ( x i^i ( A vH B ) ) C_ ( ( x i^i A ) vH B ) ) ) )
23 22 pm5.74d 
 |-  ( ( ( A e. CH /\ B e. CH ) /\ x e. CH ) -> ( ( B C_ x -> ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) ) <-> ( B C_ x -> ( x i^i ( A vH B ) ) C_ ( ( x i^i A ) vH B ) ) ) )
24 23 ralbidva 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A. x e. CH ( B C_ x -> ( ( x i^i A ) vH B ) = ( x i^i ( A vH B ) ) ) <-> A. x e. CH ( B C_ x -> ( x i^i ( A vH B ) ) C_ ( ( x i^i A ) vH B ) ) ) )
25 1 24 bitrd 
 |-  ( ( A e. CH /\ B e. CH ) -> ( A MH* B <-> A. x e. CH ( B C_ x -> ( x i^i ( A vH B ) ) C_ ( ( x i^i A ) vH B ) ) ) )