Metamath Proof Explorer


Theorem btwntriv2

Description: Betweenness always holds for the second endpoint. Theorem 3.1 of Schwabhauser p. 30. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion btwntriv2 N A 𝔼 N B 𝔼 N B Btwn A B

Proof

Step Hyp Ref Expression
1 simp1 N A 𝔼 N B 𝔼 N N
2 simp2 N A 𝔼 N B 𝔼 N A 𝔼 N
3 simp3 N A 𝔼 N B 𝔼 N B 𝔼 N
4 axsegcon N A 𝔼 N B 𝔼 N B 𝔼 N B 𝔼 N x 𝔼 N B Btwn A x B x Cgr B B
5 1 2 3 3 3 4 syl122anc N A 𝔼 N B 𝔼 N x 𝔼 N B Btwn A x B x Cgr B B
6 simpl1 N A 𝔼 N B 𝔼 N x 𝔼 N N
7 simpl3 N A 𝔼 N B 𝔼 N x 𝔼 N B 𝔼 N
8 simpr N A 𝔼 N B 𝔼 N x 𝔼 N x 𝔼 N
9 axcgrid N B 𝔼 N x 𝔼 N B 𝔼 N B x Cgr B B B = x
10 6 7 8 7 9 syl13anc N A 𝔼 N B 𝔼 N x 𝔼 N B x Cgr B B B = x
11 opeq2 B = x A B = A x
12 11 breq2d B = x B Btwn A B B Btwn A x
13 12 biimprd B = x B Btwn A x B Btwn A B
14 10 13 syl6 N A 𝔼 N B 𝔼 N x 𝔼 N B x Cgr B B B Btwn A x B Btwn A B
15 14 impd N A 𝔼 N B 𝔼 N x 𝔼 N B x Cgr B B B Btwn A x B Btwn A B
16 15 ancomsd N A 𝔼 N B 𝔼 N x 𝔼 N B Btwn A x B x Cgr B B B Btwn A B
17 16 rexlimdva N A 𝔼 N B 𝔼 N x 𝔼 N B Btwn A x B x Cgr B B B Btwn A B
18 5 17 mpd N A 𝔼 N B 𝔼 N B Btwn A B