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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 )