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Theorem btwnconn1lem14 25741
Description: Lemma for btwnconn1 25742. Final statement of the theorem when  B  =/=  C. (Contributed by Scott Fenton, 9-Oct-2013.)
Assertion
Ref Expression
btwnconn1lem14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )

Proof of Theorem btwnconn1lem14
Dummy variables  b 
c  d  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  N  e.  NN )
2 simp2l 983 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
3 simp3r 986 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
4 simp3 959 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )
5 axsegcon 25573 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. c  e.  ( EE `  N ) ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. ) )
61, 2, 3, 4, 5syl121anc 1189 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. c  e.  ( EE `  N ) ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. ) )
7 simp3l 985 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
8 axsegcon 25573 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. d  e.  ( EE `  N ) ( C  Btwn  <. A , 
d >.  /\  <. C , 
d >.Cgr <. C ,  D >. ) )
91, 2, 7, 4, 8syl121anc 1189 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. d  e.  ( EE `  N ) ( C  Btwn  <. A , 
d >.  /\  <. C , 
d >.Cgr <. C ,  D >. ) )
10 reeanv 2811 . . . 4  |-  ( E. c  e.  ( EE
`  N ) E. d  e.  ( EE
`  N ) ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) )  <->  ( E. c  e.  ( EE `  N ) ( D 
Btwn  <. A ,  c
>.  /\  <. D ,  c
>.Cgr <. C ,  D >. )  /\  E. d  e.  ( EE `  N
) ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )
116, 9, 10sylanbrc 646 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. c  e.  ( EE `  N ) E. d  e.  ( EE `  N ) ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )
1211adantr 452 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  E. c  e.  ( EE `  N ) E. d  e.  ( EE `  N ) ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )
13 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  N  e.  NN )
14 simpl2l 1010 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
15 simprl 733 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  c  e.  ( EE `  N
) )
16 simpl3l 1012 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
17 simpl2r 1011 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
18 axsegcon 25573 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  E. b  e.  ( EE `  N ) ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. ) )
1913, 14, 15, 16, 17, 18syl122anc 1193 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  E. b  e.  ( EE `  N
) ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. ) )
20 simprr 734 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  d  e.  ( EE `  N
) )
21 simpl3r 1013 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
22 axsegcon 25573 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( d  Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr <. D ,  B >. ) )
2313, 14, 20, 21, 17, 22syl122anc 1193 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N
) ( d  Btwn  <. A ,  x >.  /\ 
<. d ,  x >.Cgr <. D ,  B >. ) )
2419, 23jca 519 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  ( E. b  e.  ( EE `  N ) ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  E. x  e.  ( EE `  N
) ( d  Btwn  <. A ,  x >.  /\ 
<. d ,  x >.Cgr <. D ,  B >. ) ) )
2524adantr 452 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( E. b  e.  ( EE `  N
) ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  E. x  e.  ( EE `  N
) ( d  Btwn  <. A ,  x >.  /\ 
<. d ,  x >.Cgr <. D ,  B >. ) ) )
26 reeanv 2811 . . . . . . . 8  |-  ( E. b  e.  ( EE
`  N ) E. x  e.  ( EE
`  N ) ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) )  <->  ( E. b  e.  ( EE `  N
) ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  E. x  e.  ( EE `  N
) ( d  Btwn  <. A ,  x >.  /\ 
<. d ,  x >.Cgr <. D ,  B >. ) ) )
2725, 26sylibr 204 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  ->  E. b  e.  ( EE `  N ) E. x  e.  ( EE
`  N ) ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) )
2813, 14, 173jca 1134 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )
2928adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )
3016, 21, 153jca 1134 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) ) )
3130adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) ) )
32 simplrr 738 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  d  e.  ( EE `  N ) )
33 simprl 733 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  b  e.  ( EE `  N ) )
34 simprr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  x  e.  ( EE `  N ) )
3532, 33, 343jca 1134 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( d  e.  ( EE `  N
)  /\  b  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )
3629, 31, 353jca 1134 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
)  /\  x  e.  ( EE `  N ) ) ) )
37 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) )  ->  (
( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )
38 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) )  ->  (
( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) ) )
39 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) )  ->  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) )
4037, 38, 393jca 1134 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) )  ->  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) ) )
41 btwnconn1lem2 25729 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  c  e.  ( EE `  N ) )  /\  ( d  e.  ( EE `  N )  /\  b  e.  ( EE `  N
)  /\  x  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) ) )  ->  x  =  b )
4236, 40, 41syl2an 464 . . . . . . . . . . 11  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) ) )  ->  x  =  b )
43 opeq2 3920 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  b  ->  <. A ,  x >.  =  <. A , 
b >. )
4443breq2d 4158 . . . . . . . . . . . . . . . . 17  |-  ( x  =  b  ->  (
d  Btwn  <. A ,  x >. 
<->  d  Btwn  <. A , 
b >. ) )
45 opeq2 3920 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  b  ->  <. d ,  x >.  =  <. d ,  b >. )
4645breq1d 4156 . . . . . . . . . . . . . . . . 17  |-  ( x  =  b  ->  ( <. d ,  x >.Cgr <. D ,  B >.  <->  <. d ,  b >.Cgr <. D ,  B >. ) )
4744, 46anbi12d 692 . . . . . . . . . . . . . . . 16  |-  ( x  =  b  ->  (
( d  Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr <. D ,  B >. )  <->  ( d  Btwn  <. A ,  b >.  /\ 
<. d ,  b >.Cgr <. D ,  B >. ) ) )
4847anbi2d 685 . . . . . . . . . . . . . . 15  |-  ( x  =  b  ->  (
( ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  ( d  Btwn  <. A ,  x >.  /\ 
<. d ,  x >.Cgr <. D ,  B >. ) )  <->  ( ( c 
Btwn  <. A ,  b
>.  /\  <. c ,  b
>.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )
4948anbi2d 685 . . . . . . . . . . . . . 14  |-  ( x  =  b  ->  (
( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  ( d  Btwn  <. A ,  x >.  /\ 
<. d ,  x >.Cgr <. D ,  B >. ) ) )  <->  ( (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) ) )
5049biimpac 473 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  ( d  Btwn  <. A ,  x >.  /\ 
<. d ,  x >.Cgr <. D ,  B >. ) ) )  /\  x  =  b )  -> 
( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) ) )  /\  ( ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  ( d  Btwn  <. A ,  b >.  /\ 
<. d ,  b >.Cgr <. D ,  B >. ) ) ) )
5132, 33jca 519 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( d  e.  ( EE `  N
)  /\  b  e.  ( EE `  N ) ) )
5229, 31, 51jca32 522 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) ) )
53 simpll 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  -> 
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )
54 simplr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  -> 
( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )
55 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  -> 
( ( c  Btwn  <. A ,  b >.  /\ 
<. c ,  b >.Cgr <. C ,  B >. )  /\  ( d  Btwn  <. A ,  b >.  /\ 
<. d ,  b >.Cgr <. D ,  B >. ) ) )
5653, 54, 553jca 1134 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  -> 
( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )
57 btwnconn1lem13 25740 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N )  /\  c  e.  ( EE `  N
) )  /\  (
d  e.  ( EE
`  N )  /\  b  e.  ( EE `  N ) ) ) )  /\  ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )  ->  ( C  =  c  \/  D  =  d ) )
5852, 56, 57syl2an 464 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) ) )  ->  ( C  =  c  \/  D  =  d ) )
5958ex 424 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  b
>.  /\  <. d ,  b
>.Cgr <. D ,  B >. ) ) )  -> 
( C  =  c  \/  D  =  d ) ) )
6050, 59syl5 30 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( ( ( ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) )  /\  x  =  b )  -> 
( C  =  c  \/  D  =  d ) ) )
6160expdimp 427 . . . . . . . . . . 11  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) ) )  -> 
( x  =  b  ->  ( C  =  c  \/  D  =  d ) ) )
6242, 61mpd 15 . . . . . . . . . 10  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
b  e.  ( EE
`  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( ( ( ( A  =/=  B  /\  B  =/=  C
)  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  /\  (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) ) )  -> 
( C  =  c  \/  D  =  d ) )
6362an4s 800 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  /\  ( ( b  e.  ( EE `  N
)  /\  x  e.  ( EE `  N ) )  /\  ( ( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) ) ) )  -> 
( C  =  c  \/  D  =  d ) )
6463exp32 589 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( ( b  e.  ( EE `  N
)  /\  x  e.  ( EE `  N ) )  ->  ( (
( c  Btwn  <. A , 
b >.  /\  <. c ,  b >.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) )  ->  ( C  =  c  \/  D  =  d ) ) ) )
6564rexlimdvv 2772 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( E. b  e.  ( EE `  N
) E. x  e.  ( EE `  N
) ( ( c 
Btwn  <. A ,  b
>.  /\  <. c ,  b
>.Cgr <. C ,  B >. )  /\  ( d 
Btwn  <. A ,  x >.  /\  <. d ,  x >.Cgr
<. D ,  B >. ) )  ->  ( C  =  c  \/  D  =  d ) ) )
6627, 65mpd 15 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( C  =  c  \/  D  =  d ) )
67 orcom 377 . . . . . . 7  |-  ( ( C  =  c  \/  D  =  d )  <-> 
( D  =  d  \/  C  =  c ) )
68 simprrl 741 . . . . . . . . . 10  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  ->  C  Btwn  <. A ,  d
>. )
6968adantl 453 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  ->  C  Btwn  <. A ,  d
>. )
70 opeq2 3920 . . . . . . . . . 10  |-  ( D  =  d  ->  <. A ,  D >.  =  <. A , 
d >. )
7170breq2d 4158 . . . . . . . . 9  |-  ( D  =  d  ->  ( C  Btwn  <. A ,  D >.  <-> 
C  Btwn  <. A , 
d >. ) )
7269, 71syl5ibrcom 214 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( D  =  d  ->  C  Btwn  <. A ,  D >. ) )
73 simprll 739 . . . . . . . . . 10  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) )  ->  D  Btwn  <. A ,  c
>. )
7473adantl 453 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  ->  D  Btwn  <. A ,  c
>. )
75 opeq2 3920 . . . . . . . . . 10  |-  ( C  =  c  ->  <. A ,  C >.  =  <. A , 
c >. )
7675breq2d 4158 . . . . . . . . 9  |-  ( C  =  c  ->  ( D  Btwn  <. A ,  C >.  <-> 
D  Btwn  <. A , 
c >. ) )
7774, 76syl5ibrcom 214 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( C  =  c  ->  D  Btwn  <. A ,  C >. ) )
7872, 77orim12d 812 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( ( D  =  d  \/  C  =  c )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
7967, 78syl5bi 209 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( ( C  =  c  \/  D  =  d )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
8066, 79mpd 15 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  /\  (
( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  /\  ( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) ) ) )  -> 
( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
8180an4s 800 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  /\  ( ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) )  /\  (
( D  Btwn  <. A , 
c >.  /\  <. D , 
c >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) ) ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
8281exp32 589 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) )  ->  (
( ( D  Btwn  <. A ,  c >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  /\  ( C  Btwn  <. A ,  d >.  /\ 
<. C ,  d >.Cgr <. C ,  D >. ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) ) )
8382rexlimdvv 2772 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( E. c  e.  ( EE `  N
) E. d  e.  ( EE `  N
) ( ( D 
Btwn  <. A ,  c
>.  /\  <. D ,  c
>.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. A ,  d
>.  /\  <. C ,  d
>.Cgr <. C ,  D >. ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
8412, 83mpd 15 1  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   <.cop 3753   class class class wbr 4146   ` cfv 5387   NNcn 9925   EEcee 25534    Btwn cbtwn 25535  Cgrccgr 25536
This theorem is referenced by:  btwnconn1  25742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-ee 25537  df-btwn 25538  df-cgr 25539  df-ofs 25624  df-ifs 25680  df-cgr3 25681  df-colinear 25682  df-fs 25683
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