Metamath Proof Explorer

Theorem colineartriv2

Description: Trivial case of colinearity. (Contributed by Scott Fenton, 18-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion colineartriv2 
|- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A Colinear <. B , B >. )

Proof

Step Hyp Ref Expression
1 btwntriv1 
 |-  ( ( N e. NN /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) -> B Btwn <. B , A >. )
2 1 3mix2d 
 |-  ( ( N e. NN /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) -> ( A Btwn <. B , B >. \/ B Btwn <. B , A >. \/ B Btwn <. A , B >. ) )
3 2 3com23 
 |-  ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A Btwn <. B , B >. \/ B Btwn <. B , A >. \/ B Btwn <. A , B >. ) )
4 simp1 
 |-  ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> N e. NN )
5 simp2 
 |-  ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A e. ( EE ` N ) )
6 simp3 
 |-  ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> B e. ( EE ` N ) )
7 brcolinear 
 |-  ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Colinear <. B , B >. <-> ( A Btwn <. B , B >. \/ B Btwn <. B , A >. \/ B Btwn <. A , B >. ) ) )
8 4 5 6 6 7 syl13anc 
 |-  ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A Colinear <. B , B >. <-> ( A Btwn <. B , B >. \/ B Btwn <. B , A >. \/ B Btwn <. A , B >. ) ) )
9 3 8 mpbird 
 |-  ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A Colinear <. B , B >. )