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Metamath Proof Explorer

Theorem brdifun

Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014)

Ref Expression
Hypothesis swoer.1 
|- R = ( ( X X. X ) \ ( .< u. `' .< ) )
Assertion brdifun 
|- ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )

Proof

Step Hyp Ref Expression
1 swoer.1 
 |-  R = ( ( X X. X ) \ ( .< u. `' .< ) )
2 opelxpi 
 |-  ( ( A e. X /\ B e. X ) -> <. A , B >. e. ( X X. X ) )
3 df-br 
 |-  ( A ( X X. X ) B <-> <. A , B >. e. ( X X. X ) )
4 2 3 sylibr 
 |-  ( ( A e. X /\ B e. X ) -> A ( X X. X ) B )
5 1 breqi 
 |-  ( A R B <-> A ( ( X X. X ) \ ( .< u. `' .< ) ) B )
6 brdif 
 |-  ( A ( ( X X. X ) \ ( .< u. `' .< ) ) B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
7 5 6 bitri 
 |-  ( A R B <-> ( A ( X X. X ) B /\ -. A ( .< u. `' .< ) B ) )
8 7 baib 
 |-  ( A ( X X. X ) B -> ( A R B <-> -. A ( .< u. `' .< ) B ) )
9 4 8 syl 
 |-  ( ( A e. X /\ B e. X ) -> ( A R B <-> -. A ( .< u. `' .< ) B ) )
10 brun 
 |-  ( A ( .< u. `' .< ) B <-> ( A .< B \/ A `' .< B ) )
11 brcnvg 
 |-  ( ( A e. X /\ B e. X ) -> ( A `' .< B <-> B .< A ) )
12 11 orbi2d 
 |-  ( ( A e. X /\ B e. X ) -> ( ( A .< B \/ A `' .< B ) <-> ( A .< B \/ B .< A ) ) )
13 10 12 bitrid 
 |-  ( ( A e. X /\ B e. X ) -> ( A ( .< u. `' .< ) B <-> ( A .< B \/ B .< A ) ) )
14 13 notbid 
 |-  ( ( A e. X /\ B e. X ) -> ( -. A ( .< u. `' .< ) B <-> -. ( A .< B \/ B .< A ) ) )
15 9 14 bitrd 
 |-  ( ( A e. X /\ B e. X ) -> ( A R B <-> -. ( A .< B \/ B .< A ) ) )