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Theorem btwnconn1lem7 24788
Description: Lemma for btwnconn1 24796. 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 1050 . . . . 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 451 . . . 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 1025 . . . . 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 451 . . . 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 1023 . . . . 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 451 . . . 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 1132 . . 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 985 . . . 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 988 . . . 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 989 . . . 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 990 . . . 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 991 . . . 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 961 . . . . . 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 3813 . . . . . . . . . . . 12  |-  ( C  =  d  ->  <. C ,  C >.  =  <. C , 
d >. )
1514breq1d 4049 . . . . . . . . . . 11  |-  ( C  =  d  ->  ( <. C ,  C >.Cgr <. C ,  D >.  <->  <. C ,  d >.Cgr <. C ,  D >. ) )
16153anbi2d 1257 . . . . . . . . . 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 473 . . . . . . . . 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 956 . . . . . . . . . . . . 13  |-  ( ( C  =/=  c  /\  <. C ,  C >.Cgr <. C ,  D >.  /\ 
<. D ,  c >.Cgr <. C ,  D >. )  ->  <. C ,  C >.Cgr
<. C ,  D >. )
19 simp1 955 . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . 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 982 . . . . . . . . . . . . . 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 24698 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 28 . . . . . . . . . . . 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 418 . . . . . . . . . . 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 3812 . . . . . . . . . . . . . . . 16  |-  ( C  =  D  ->  <. C , 
c >.  =  <. D , 
c >. )
27 opeq2 3813 . . . . . . . . . . . . . . . 16  |-  ( C  =  D  ->  <. C ,  C >.  =  <. C ,  D >. )
2826, 27breq12d 4052 . . . . . . . . . . . . . . 15  |-  ( C  =  D  ->  ( <. C ,  c >.Cgr <. C ,  C >.  <->  <. D ,  c >.Cgr <. C ,  D >. ) )
2928biimparc 473 . . . . . . . . . . . . . 14  |-  ( (
<. D ,  c >.Cgr <. C ,  D >.  /\  C  =  D )  ->  <. C ,  c
>.Cgr <. C ,  C >. )
30 simp3l 983 . . . . . . . . . . . . . . 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 24616 . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . 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 28 . . . . . . . . . . . . 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 426 . . . . . . . . . . . 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 1122 . . . . . . . . . . 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 14 . . . . . . . . . 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 423 . . . . . . . . 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 28 . . . . . . . 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 426 . . . . . . 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 2497 . . . . . 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 14 . . . . 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 423 . . . 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 1191 . . 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 28 . 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 418 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   <.cop 3656   class class class wbr 4039   ` cfv 5271   NNcn 9762   EEcee 24588    Btwn cbtwn 24589  Cgrccgr 24590
This theorem is referenced by:  btwnconn1lem8  24789  btwnconn1lem12  24793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-ee 24591  df-cgr 24593
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