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Theorem outsideofeq 28295
Description: Uniqueness law for OutsideOf. Analog of segconeq 28175. (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 28287 . . . . 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 1221 . . . 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 28287 . . . . 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 1221 . . . 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 28160 . . . . 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 28269 . . . . . . . . . 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 1228 . . . . . . . . 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 1318 . . . . . . 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 28191 . . . . . . . 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 28251 . . . . . . 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 28191 . . . . . . . . 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 28156 . . . . . . . . 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 28251 . . . . . . . 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 2459 . . . . . . 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 28156 . . . . . . . . . . 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 28251 . . . . . . . . . 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 2459 . . . . . . . . 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 28251 . . . . . . . . 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 2719 . . . . . . . . 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 28266 . . . . . . . . . . 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 1228 . . . . . . . . . 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 1318 . . . . . . . 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 1370    e. wcel 1758    =/= wne 2644   <.cop 3981   class class class wbr 4390   ` cfv 5516   NNcn 10423   EEcee 23269    Btwn cbtwn 23270  Cgrccgr 23271  OutsideOfcoutsideof 28284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-sum 13266  df-ee 23272  df-btwn 23273  df-cgr 23274  df-ofs 28148  df-colinear 28204  df-ifs 28205  df-cgr3 28206  df-fs 28207  df-outsideof 28285
This theorem is referenced by:  outsideofeu  28296  outsidele  28297
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