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Theorem btwnconn1lem7 29948
Description: Lemma for btwnconn1 29956. 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 1091 . . . . 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 465 . . . 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 1066 . . . . 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 465 . . . 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 1064 . . . . 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 465 . . . 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 1176 . . 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 1026 . . . 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 1029 . . . 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 1030 . . . 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 1031 . . . 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 1032 . . . 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 1002 . . . . . 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 4220 . . . . . . . . . . . 12  |-  ( C  =  d  ->  <. C ,  C >.  =  <. C , 
d >. )
1514breq1d 4466 . . . . . . . . . . 11  |-  ( C  =  d  ->  ( <. C ,  C >.Cgr <. C ,  D >.  <->  <. C ,  d >.Cgr <. C ,  D >. ) )
16153anbi2d 1304 . . . . . . . . . 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 487 . . . . . . . . 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 997 . . . . . . . . . . . . 13  |-  ( ( C  =/=  c  /\  <. C ,  C >.Cgr <. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  ->  <. C ,  C >.Cgr
<. C ,  D >. )
19 simp1 996 . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . 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 1023 . . . . . . . . . . . . . 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 29858 . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . 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 32 . . . . . . . . . . . 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 429 . . . . . . . . . . 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 4219 . . . . . . . . . . . . . . . 16  |-  ( C  =  D  ->  <. C , 
c >.  =  <. D , 
c >. )
27 opeq2 4220 . . . . . . . . . . . . . . . 16  |-  ( C  =  D  ->  <. C ,  C >.  =  <. C ,  D >. )
2826, 27breq12d 4469 . . . . . . . . . . . . . . 15  |-  ( C  =  D  ->  ( <. C ,  c >.Cgr <. C ,  C >.  <->  <. D ,  c >.Cgr <. C ,  D >. ) )
2928biimparc 487 . . . . . . . . . . . . . 14  |-  ( (
<. D ,  c >.Cgr <. C ,  D >.  /\  C  =  D )  ->  <. C ,  c
>.Cgr <. C ,  C >. )
30 simp3l 1024 . . . . . . . . . . . . . . 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 24346 . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . 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 437 . . . . . . . . . . . 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 1163 . . . . . . . . . . 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 434 . . . . . . . . 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 32 . . . . . . . 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 437 . . . . . . 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 2681 . . . . . 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 434 . . . 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 1237 . . 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 32 . 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 429 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 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   <.cop 4038   class class class wbr 4456   ` cfv 5594   NNcn 10556   EEcee 24318    Btwn cbtwn 24319  Cgrccgr 24320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-ee 24321  df-cgr 24323
This theorem is referenced by:  btwnconn1lem8  29949  btwnconn1lem12  29953
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