Metamath Proof Explorer


Theorem sdomdomtr

Description: Transitivity of strict dominance and dominance. Theorem 22(iii) of Suppes p. 97. (Contributed by NM, 26-Oct-2003) (Revised by Mario Carneiro, 26-Apr-2015)

Ref Expression
Assertion sdomdomtr
|- ( ( A ~< B /\ B ~<_ C ) -> A ~< C )

Proof

Step Hyp Ref Expression
1 sdomdom
 |-  ( A ~< B -> A ~<_ B )
2 domtr
 |-  ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
3 1 2 sylan
 |-  ( ( A ~< B /\ B ~<_ C ) -> A ~<_ C )
4 simpl
 |-  ( ( A ~< B /\ B ~<_ C ) -> A ~< B )
5 simpr
 |-  ( ( A ~< B /\ B ~<_ C ) -> B ~<_ C )
6 ensym
 |-  ( A ~~ C -> C ~~ A )
7 domentr
 |-  ( ( B ~<_ C /\ C ~~ A ) -> B ~<_ A )
8 5 6 7 syl2an
 |-  ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> B ~<_ A )
9 domnsym
 |-  ( B ~<_ A -> -. A ~< B )
10 8 9 syl
 |-  ( ( ( A ~< B /\ B ~<_ C ) /\ A ~~ C ) -> -. A ~< B )
11 10 ex
 |-  ( ( A ~< B /\ B ~<_ C ) -> ( A ~~ C -> -. A ~< B ) )
12 4 11 mt2d
 |-  ( ( A ~< B /\ B ~<_ C ) -> -. A ~~ C )
13 brsdom
 |-  ( A ~< C <-> ( A ~<_ C /\ -. A ~~ C ) )
14 3 12 13 sylanbrc
 |-  ( ( A ~< B /\ B ~<_ C ) -> A ~< C )