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Theorem outsideofeq 28130
Description: Uniqueness law for OutsideOf. Analog of segconeq 28010. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideofeq  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( ( AOutsideOf <. X ,  R >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( AOutsideOf <. Y ,  R >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) )  ->  X  =  Y ) )

Proof of Theorem outsideofeq
StepHypRef Expression
1 simp1 988 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  N  e.  NN )
2 simp21 1021 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
3 simp32 1025 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  X  e.  ( EE `  N
) )
4 simp22 1022 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  R  e.  ( EE `  N
) )
5 broutsideof2 28122 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  X  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) ) )  -> 
( AOutsideOf <. X ,  R >.  <-> 
( X  =/=  A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) ) ) )
61, 2, 3, 4, 5syl13anc 1220 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  ( AOutsideOf
<. X ,  R >.  <->  ( X  =/=  A  /\  R  =/=  A  /\  ( X 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) ) ) )
76anbi1d 704 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( AOutsideOf <. X ,  R >.  /\  <. A ,  X >.Cgr
<. B ,  C >. )  <-> 
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. ) ) )
8 simp33 1026 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  Y  e.  ( EE `  N
) )
9 broutsideof2 28122 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  Y  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) ) )  -> 
( AOutsideOf <. Y ,  R >.  <-> 
( Y  =/=  A  /\  R  =/=  A  /\  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) ) ) )
101, 2, 8, 4, 9syl13anc 1220 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  ( AOutsideOf
<. Y ,  R >.  <->  ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) ) ) )
1110anbi1d 704 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( AOutsideOf <. Y ,  R >.  /\  <. A ,  Y >.Cgr
<. B ,  C >. )  <-> 
( ( Y  =/= 
A  /\  R  =/=  A  /\  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )
127, 11anbi12d 710 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( ( AOutsideOf <. X ,  R >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( AOutsideOf <. Y ,  R >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) )  <->  ( ( ( X  =/=  A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) ) )
13 simpll3 1029 . . . . . . 7  |-  ( ( ( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )
14 simprl3 1035 . . . . . . 7  |-  ( ( ( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )
1513, 14jca 532 . . . . . 6  |-  ( ( ( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  ( ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. )  /\  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) ) )
1615adantl 466 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  (
( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. )  /\  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) ) )
17 simpll2 1028 . . . . . 6  |-  ( ( ( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  R  =/=  A )
1817adantl 466 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  R  =/=  A )
19 simp23 1023 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
20 simp31 1024 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
21 simprlr 762 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  X >.Cgr <. B ,  C >. )
22 simprrr 764 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  Y >.Cgr <. B ,  C >. )
231, 2, 3, 2, 8, 19, 20, 21, 22cgrtr3and 27995 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
2416, 18, 23jca32 535 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  (
( ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. )  /\  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr
<. A ,  Y >. ) ) )
25 simprll 761 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  X  Btwn  <. A ,  R >. )
26 simprlr 762 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  Y  Btwn  <. A ,  R >. )
27 simprrr 764 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
28 midofsegid 28104 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( X  e.  ( EE `  N )  /\  Y  e.  ( EE `  N ) ) )  ->  ( ( X 
Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >.  /\  <. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) )
291, 2, 4, 3, 8, 28syl122anc 1227 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >.  /\ 
<. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) )
3029adantr 465 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  -> 
( ( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >.  /\  <. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) )
3125, 26, 27, 30mp3and 1317 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  X  =  Y )
3231exp32 605 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( X  Btwn  <. A ,  R >.  /\  Y  Btwn  <. A ,  R >. )  ->  ( ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) ) )
33 simprlr 762 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  Y  Btwn  <. A ,  R >. )
34 simprll 761 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  R  Btwn  <. A ,  X >. )
351, 2, 8, 4, 3, 33, 34btwnexchand 28026 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  Y  Btwn  <. A ,  X >. )
36 simprrr 764 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
371, 2, 3, 8, 35, 36endofsegidand 28086 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  Y  Btwn  <. A ,  R >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  X  =  Y )
3837exp32 605 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( R  Btwn  <. A ,  X >.  /\  Y  Btwn  <. A ,  R >. )  ->  ( ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) ) )
39 simprll 761 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  X  Btwn  <. A ,  R >. )
40 simprlr 762 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  R  Btwn  <. A ,  Y >. )
411, 2, 3, 4, 8, 39, 40btwnexchand 28026 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  X  Btwn  <. A ,  Y >. )
42 simprrr 764 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
431, 2, 3, 2, 8, 42cgrcomand 27991 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  <. A ,  Y >.Cgr <. A ,  X >. )
441, 2, 8, 3, 41, 43endofsegidand 28086 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  Y  =  X )
4544eqcomd 2443 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  X  =  Y )
4645exp32 605 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( X  Btwn  <. A ,  R >.  /\  R  Btwn  <. A ,  Y >. )  ->  ( ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) ) )
47 simprr 756 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  X  Btwn  <. A ,  Y >. ) )  ->  X  Btwn  <. A ,  Y >. )
48 simplrr 760 . . . . . . . . . . . . 13  |-  ( ( ( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  X  Btwn  <. A ,  Y >. )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
4948adantl 466 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  X  Btwn  <. A ,  Y >. ) )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
501, 2, 3, 2, 8, 49cgrcomand 27991 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  X  Btwn  <. A ,  Y >. ) )  ->  <. A ,  Y >.Cgr <. A ,  X >. )
511, 2, 8, 3, 47, 50endofsegidand 28086 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  X  Btwn  <. A ,  Y >. ) )  ->  Y  =  X )
5251eqcomd 2443 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  X  Btwn  <. A ,  Y >. ) )  ->  X  =  Y )
5352expr 615 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  -> 
( X  Btwn  <. A ,  Y >.  ->  X  =  Y ) )
54 simprr 756 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  Y  Btwn  <. A ,  X >. ) )  ->  Y  Btwn  <. A ,  X >. )
55 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  Y  Btwn  <. A ,  X >. )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
5655adantl 466 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  Y  Btwn  <. A ,  X >. ) )  ->  <. A ,  X >.Cgr <. A ,  Y >. )
571, 2, 3, 8, 54, 56endofsegidand 28086 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) )  /\  Y  Btwn  <. A ,  X >. ) )  ->  X  =  Y )
5857expr 615 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  -> 
( Y  Btwn  <. A ,  X >.  ->  X  =  Y ) )
59 simprrl 763 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  R  =/=  A )
6059necomd 2690 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  A  =/=  R )
61 simprll 761 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  R  Btwn  <. A ,  X >. )
62 simprlr 762 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  R  Btwn  <. A ,  Y >. )
63 btwnconn1 28101 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( X  e.  ( EE `  N )  /\  Y  e.  ( EE `  N ) ) )  ->  ( ( A  =/=  R  /\  R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  ->  ( X  Btwn  <. A ,  Y >.  \/  Y  Btwn  <. A ,  X >. ) ) )
641, 2, 4, 3, 8, 63syl122anc 1227 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( A  =/=  R  /\  R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  ->  ( X  Btwn  <. A ,  Y >.  \/  Y  Btwn  <. A ,  X >. ) ) )
6564adantr 465 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  -> 
( ( A  =/= 
R  /\  R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  ->  ( X  Btwn  <. A ,  Y >.  \/  Y  Btwn  <. A ,  X >. ) ) )
6660, 61, 62, 65mp3and 1317 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  -> 
( X  Btwn  <. A ,  Y >.  \/  Y  Btwn  <. A ,  X >. ) )
6753, 58, 66mpjaod 381 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  /\  ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. ) ) )  ->  X  =  Y )
6867exp32 605 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( R  Btwn  <. A ,  X >.  /\  R  Btwn  <. A ,  Y >. )  ->  ( ( R  =/=  A  /\  <. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) ) )
6932, 38, 46, 68ccased 938 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. )  /\  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  -> 
( ( R  =/= 
A  /\  <. A ,  X >.Cgr <. A ,  Y >. )  ->  X  =  Y ) ) )
7069imp32 433 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. )  /\  ( Y  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  ( R  =/=  A  /\  <. A ,  X >.Cgr
<. A ,  Y >. ) ) )  ->  X  =  Y )
7124, 70syldan 470 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  /\  (
( ( X  =/= 
A  /\  R  =/=  A  /\  ( X  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) ) )  ->  X  =  Y )
7271ex 434 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( ( ( X  =/=  A  /\  R  =/=  A  /\  ( X 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  X >. ) )  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( ( Y  =/=  A  /\  R  =/=  A  /\  ( Y 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  Y >. ) )  /\  <. A ,  Y >.Cgr <. B ,  C >. ) )  ->  X  =  Y ) )
7312, 72sylbid 215 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  X  e.  ( EE `  N
)  /\  Y  e.  ( EE `  N ) ) )  ->  (
( ( AOutsideOf <. X ,  R >.  /\  <. A ,  X >.Cgr <. B ,  C >. )  /\  ( AOutsideOf <. Y ,  R >.  /\ 
<. A ,  Y >.Cgr <. B ,  C >. ) )  ->  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   <.cop 3878   class class class wbr 4287   ` cfv 5413   NNcn 10314   EEcee 23102    Btwn cbtwn 23103  Cgrccgr 23104  OutsideOfcoutsideof 28119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-ee 23105  df-btwn 23106  df-cgr 23107  df-ofs 27983  df-colinear 28039  df-ifs 28040  df-cgr3 28041  df-fs 28042  df-outsideof 28120
This theorem is referenced by:  outsideofeu  28131  outsidele  28132
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