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Theorem outsideofeq 24825
Description: Uniqueness law for OutsideOf. Analog of segconeq 24705. (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 955 . . . . 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 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 ) ) )  ->  A  e.  ( EE `  N
) )
3 simp32 992 . . . . 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 989 . . . . 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 24817 . . . . 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 1184 . . . 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 685 . . 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 993 . . . . 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 24817 . . . . 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 1184 . . . 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 685 . . 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 691 . 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 996 . . . . . . 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 1002 . . . . . . 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 518 . . . . . 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 452 . . . . 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 995 . . . . . 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 452 . . . . 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 990 . . . . . 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 991 . . . . . 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 739 . . . . . 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 741 . . . . . 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 24690 . . . . 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 521 . . . 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 738 . . . . . . . 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 739 . . . . . . . 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 741 . . . . . . . 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 24799 . . . . . . . . . 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 1191 . . . . . . . . 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 451 . . . . . . . 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 1280 . . . . . . 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 588 . . . . . 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 739 . . . . . . . . 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 738 . . . . . . . . 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 24721 . . . . . . . 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 741 . . . . . . . 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 24781 . . . . . . 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 588 . . . . . 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 738 . . . . . . . . . 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 739 . . . . . . . . . 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 24721 . . . . . . . . 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 741 . . . . . . . . . 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 24686 . . . . . . . . 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 24781 . . . . . . . 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 2301 . . . . . . 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 588 . . . . . 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 733 . . . . . . . . . . 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 737 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 24686 . . . . . . . . . . 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 24781 . . . . . . . . . 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 2301 . . . . . . . . 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 598 . . . . . . . 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 733 . . . . . . . . . 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 737 . . . . . . . . . . 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 452 . . . . . . . . . 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 24781 . . . . . . . . 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 598 . . . . . . . 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 740 . . . . . . . . . 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 2542 . . . . . . . . 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 738 . . . . . . . . 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 739 . . . . . . . . 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 24796 . . . . . . . . . . 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 1191 . . . . . . . . . 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 451 . . . . . . . . 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 1280 . . . . . . . 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 370 . . . . . . 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 588 . . . . . 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 913 . . . . 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 422 . . . 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 456 . . 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 423 . 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 206 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 set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ 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  OutsideOfcoutsideof 24814
This theorem is referenced by:  outsideofeu  24826  outsidele  24827
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-icc 10679  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-btwn 24592  df-cgr 24593  df-ofs 24678  df-ifs 24734  df-cgr3 24735  df-colinear 24736  df-fs 24737  df-outsideof 24815
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