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Theorem outsideofeq 30897
Description: Uniqueness law for OutsideOf. Analogue of segconeq 30777. (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 1008 . . . . 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 1041 . . . . 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 1045 . . . . 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 1042 . . . . 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 30889 . . . . 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 1270 . . . 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 711 . . 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 1046 . . . . 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 30889 . . . . 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 1270 . . . 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 711 . . 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 717 . 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 1049 . . . . . . 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 1055 . . . . . . 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 535 . . . . . 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 468 . . . . 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 1048 . . . . . 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 468 . . . . 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 1043 . . . . . 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 1044 . . . . . 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 773 . . . . . 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 775 . . . . . 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 30762 . . . . 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 538 . . . 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 772 . . . . . . . 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 773 . . . . . . . 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 775 . . . . . . . 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 30871 . . . . . . . . . 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 1277 . . . . . . . . 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 467 . . . . . . . 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 1367 . . . . . . 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 610 . . . . . 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 773 . . . . . . . . 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 772 . . . . . . . . 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 30793 . . . . . . . 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 775 . . . . . . . 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 30853 . . . . . . 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 610 . . . . . 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 772 . . . . . . . . . 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 773 . . . . . . . . . 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 30793 . . . . . . . . 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 775 . . . . . . . . . 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 30758 . . . . . . . . 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 30853 . . . . . . . 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 2457 . . . . . . 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 610 . . . . . 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 766 . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 468 . . . . . . . . . . . 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 30758 . . . . . . . . . . 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 30853 . . . . . . . . . 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 2457 . . . . . . . . 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 620 . . . . . . . 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 766 . . . . . . . . . 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 771 . . . . . . . . . . 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 468 . . . . . . . . . 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 30853 . . . . . . . . 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 620 . . . . . . . 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 774 . . . . . . . . . 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 2679 . . . . . . . . 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 772 . . . . . . . . 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 773 . . . . . . . . 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 30868 . . . . . . . . . . 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 1277 . . . . . . . . . 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 467 . . . . . . . . 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 1367 . . . . . . . 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 383 . . . . . . 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 610 . . . . . 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 958 . . . . 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 435 . . . 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 473 . . 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 436 . 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 219 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 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   <.cop 3974   class class class wbr 4402   ` cfv 5582   NNcn 10609   EEcee 24918    Btwn cbtwn 24919  Cgrccgr 24920  OutsideOfcoutsideof 30886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-ee 24921  df-btwn 24922  df-cgr 24923  df-ofs 30750  df-colinear 30806  df-ifs 30807  df-cgr3 30808  df-fs 30809  df-outsideof 30887
This theorem is referenced by:  outsideofeu  30898  outsidele  30899
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