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Theorem btwnconn1lem7 30642
Description: Lemma for btwnconn1 30650. Under our assumptions,  C and  d are distinct. (Contributed by Scott Fenton, 8-Oct-2013.)
Assertion
Ref Expression
btwnconn1lem7  |-  ( ( ( ( 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
)  /\  E  e.  ( EE `  N ) ) )  /\  (
( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  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 >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) ) )  ->  C  =/=  d )

Proof of Theorem btwnconn1lem7
StepHypRef Expression
1 simp1l3 1100 . . . . 5  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  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 )
21adantr 466 . . . 4  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  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 >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) )  ->  C  =/=  c )
3 simp2rr 1075 . . . . 5  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  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 ,  d >.Cgr <. C ,  D >. )
43adantr 466 . . . 4  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  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 >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) )  ->  <. C , 
d >.Cgr <. C ,  D >. )
5 simp2lr 1073 . . . . 5  |-  ( ( ( ( A  =/= 
B  /\  B  =/=  C  /\  C  =/=  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 ,  c >.Cgr <. C ,  D >. )
65adantr 466 . . . 4  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  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 >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) )  ->  <. D , 
c >.Cgr <. C ,  D >. )
72, 4, 63jca 1185 . . 3  |-  ( ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  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 >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) )  ->  ( C  =/=  c  /\  <. C , 
d >.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. ) )
8 simp11 1035 . . . 4  |-  ( ( ( 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 )  /\  E  e.  ( EE `  N
) ) )  ->  N  e.  NN )
9 simp21 1038 . . . 4  |-  ( ( ( 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 )  /\  E  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
10 simp22 1039 . . . 4  |-  ( ( ( 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 )  /\  E  e.  ( EE `  N
) ) )  ->  D  e.  ( EE `  N ) )
11 simp23 1040 . . . 4  |-  ( ( ( 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 )  /\  E  e.  ( EE `  N
) ) )  -> 
c  e.  ( EE
`  N ) )
12 simp31 1041 . . . 4  |-  ( ( ( 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 )  /\  E  e.  ( EE `  N
) ) )  -> 
d  e.  ( EE
`  N ) )
13 simpr1 1011 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  ( C  =/=  c  /\  <. C ,  d
>.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. ) )  ->  C  =/=  c )
14 opeq2 4182 . . . . . . . . . . . 12  |-  ( C  =  d  ->  <. C ,  C >.  =  <. C , 
d >. )
1514breq1d 4427 . . . . . . . . . . 11  |-  ( C  =  d  ->  ( <. C ,  C >.Cgr <. C ,  D >.  <->  <. C ,  d >.Cgr <. C ,  D >. ) )
16153anbi2d 1340 . . . . . . . . . 10  |-  ( C  =  d  ->  (
( C  =/=  c  /\  <. C ,  C >.Cgr
<. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  <-> 
( C  =/=  c  /\  <. C ,  d
>.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. ) ) )
1716biimparc 489 . . . . . . . . 9  |-  ( ( ( C  =/=  c  /\  <. C ,  d
>.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. )  /\  C  =  d )  ->  ( C  =/=  c  /\  <. C ,  C >.Cgr <. C ,  D >.  /\  <. D , 
c >.Cgr <. C ,  D >. ) )
18 simp2 1006 . . . . . . . . . . . . 13  |-  ( ( C  =/=  c  /\  <. C ,  C >.Cgr <. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  ->  <. C ,  C >.Cgr
<. C ,  D >. )
19 simp1 1005 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  N  e.  NN )
20 simp2l 1031 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
21 simp2r 1032 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
22 cgrid2 30552 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. C ,  C >.Cgr
<. C ,  D >.  ->  C  =  D )
)
2319, 20, 20, 21, 22syl13anc 1266 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  ( <. C ,  C >.Cgr <. C ,  D >.  ->  C  =  D )
)
2418, 23syl5 33 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  (
( C  =/=  c  /\  <. C ,  C >.Cgr
<. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  ->  C  =  D ) )
2524imp 430 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  ( C  =/=  c  /\  <. C ,  C >.Cgr
<. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. ) )  ->  C  =  D )
26 opeq1 4181 . . . . . . . . . . . . . . . 16  |-  ( C  =  D  ->  <. C , 
c >.  =  <. D , 
c >. )
27 opeq2 4182 . . . . . . . . . . . . . . . 16  |-  ( C  =  D  ->  <. C ,  C >.  =  <. C ,  D >. )
2826, 27breq12d 4430 . . . . . . . . . . . . . . 15  |-  ( C  =  D  ->  ( <. C ,  c >.Cgr <. C ,  C >.  <->  <. D ,  c >.Cgr <. C ,  D >. ) )
2928biimparc 489 . . . . . . . . . . . . . 14  |-  ( (
<. D ,  c >.Cgr <. C ,  D >.  /\  C  =  D )  ->  <. C ,  c
>.Cgr <. C ,  C >. )
30 simp3l 1033 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  c  e.  ( EE `  N
) )
31 axcgrid 24789 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  c  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. C ,  c
>.Cgr <. C ,  C >.  ->  C  =  c ) )
3219, 20, 30, 20, 31syl13anc 1266 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  ( <. C ,  c >.Cgr <. C ,  C >.  ->  C  =  c )
)
3329, 32syl5 33 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  (
( <. D ,  c
>.Cgr <. C ,  D >.  /\  C  =  D )  ->  C  =  c ) )
3433expdimp 438 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  <. D ,  c >.Cgr <. C ,  D >. )  ->  ( C  =  D  ->  C  =  c ) )
35343ad2antr3 1172 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  ( C  =/=  c  /\  <. C ,  C >.Cgr
<. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. ) )  ->  ( C  =  D  ->  C  =  c ) )
3625, 35mpd 15 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  ( C  =/=  c  /\  <. C ,  C >.Cgr
<. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. ) )  ->  C  =  c )
3736ex 435 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  (
( C  =/=  c  /\  <. C ,  C >.Cgr
<. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  ->  C  =  c ) )
3817, 37syl5 33 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  (
( ( C  =/=  c  /\  <. C , 
d >.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. )  /\  C  =  d )  ->  C  =  c ) )
3938expdimp 438 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  ( C  =/=  c  /\  <. C ,  d
>.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. ) )  ->  ( C  =  d  ->  C  =  c ) )
4039necon3d 2646 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  ( C  =/=  c  /\  <. C ,  d
>.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. ) )  ->  ( C  =/=  c  ->  C  =/=  d ) )
4113, 40mpd 15 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N
) ) )  /\  ( C  =/=  c  /\  <. C ,  d
>.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. ) )  ->  C  =/=  d )
4241ex 435 . . . 4  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( c  e.  ( EE `  N )  /\  d  e.  ( EE `  N ) ) )  ->  (
( C  =/=  c  /\  <. C ,  d
>.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. )  ->  C  =/=  d ) )
438, 9, 10, 11, 12, 42syl122anc 1273 . . 3  |-  ( ( ( 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 )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( C  =/=  c  /\  <. C , 
d >.Cgr <. C ,  D >.  /\  <. D ,  c
>.Cgr <. C ,  D >. )  ->  C  =/=  d ) )
447, 43syl5 33 . 2  |-  ( ( ( 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 )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( ( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  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 >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) )  ->  C  =/=  d ) )
4544imp 430 1  |-  ( ( ( ( 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
)  /\  E  e.  ( EE `  N ) ) )  /\  (
( ( ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  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 >. ) ) )  /\  ( E  Btwn  <. C , 
c >.  /\  E  Btwn  <. D ,  d >. ) ) )  ->  C  =/=  d )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   <.cop 3999   class class class wbr 4417   ` cfv 5592   NNcn 10598   EEcee 24761    Btwn cbtwn 24762  Cgrccgr 24763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-ico 11630  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-sum 13720  df-ee 24764  df-cgr 24766
This theorem is referenced by:  btwnconn1lem8  30643  btwnconn1lem12  30647
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