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Theorem btwnoutside 30450
Description: A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
btwnoutside  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P )  /\  P  Btwn  <. A ,  C >. )  ->  ( P  Btwn  <. B ,  C >.  <-> 
POutsideOf <. A ,  B >. ) ) )

Proof of Theorem btwnoutside
StepHypRef Expression
1 df-3an 976 . . . . . 6  |-  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. )  <->  ( (
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. )  /\  P  Btwn  <. B ,  C >. ) )
2 simpr11 1081 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  ->  A  =/=  P )
3 simpr12 1082 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  ->  B  =/=  P )
4 simpr13 1083 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  ->  C  =/=  P )
5 simp1 997 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  N  e.  NN )
6 simp3r 1026 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  P  e.  ( EE `  N ) )
7 simp2l 1023 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
8 simp3l 1025 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
9 simpr2 1004 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  ->  P  Btwn  <. A ,  C >. )
105, 6, 7, 8, 9btwncomand 30340 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  ->  P  Btwn  <. C ,  A >. )
11 simp2r 1024 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N ) )
12 simpr3 1005 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  ->  P  Btwn  <. B ,  C >. )
135, 6, 11, 8, 12btwncomand 30340 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  ->  P  Btwn  <. C ,  B >. )
14 btwnconn2 30427 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( ( C  =/=  P  /\  P  Btwn  <. C ,  A >.  /\  P  Btwn  <. C ,  B >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
15143com23 1203 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( C  =/=  P  /\  P  Btwn  <. C ,  A >.  /\  P  Btwn  <. C ,  B >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
1615adantr 463 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  -> 
( ( C  =/= 
P  /\  P  Btwn  <. C ,  A >.  /\  P  Btwn  <. C ,  B >. )  ->  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
174, 10, 13, 16mp3and 1329 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  -> 
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
182, 3, 173jca 1177 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  P  Btwn  <. B ,  C >. ) )  -> 
( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
191, 18sylan2br 474 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  P  Btwn  <. A ,  C >. )  /\  P  Btwn  <. B ,  C >. ) )  ->  ( A  =/=  P  /\  B  =/= 
P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) )
2019expr 613 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( P  Btwn  <. B ,  C >.  ->  ( A  =/= 
P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
21 simp3 999 . . . . 5  |-  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  -> 
( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )
22 df-3an 976 . . . . . . . 8  |-  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  A  Btwn  <. P ,  B >. )  <->  ( (
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. )  /\  A  Btwn  <. P ,  B >. ) )
23 simpr11 1081 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  A  Btwn  <. P ,  B >. ) )  ->  A  =/=  P )
24 simpr3 1005 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  A  Btwn  <. P ,  B >. ) )  ->  A  Btwn  <. P ,  B >. )
255, 7, 6, 11, 24btwncomand 30340 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  A  Btwn  <. P ,  B >. ) )  ->  A  Btwn  <. B ,  P >. )
26 simpr2 1004 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  A  Btwn  <. P ,  B >. ) )  ->  P  Btwn  <. A ,  C >. )
27 btwnouttr2 30347 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( P  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( A  =/=  P  /\  A  Btwn  <. B ,  P >.  /\  P  Btwn  <. A ,  C >. )  ->  P  Btwn  <. B ,  C >. ) )
285, 11, 7, 6, 8, 27syl122anc 1239 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( A  =/=  P  /\  A  Btwn  <. B ,  P >.  /\  P  Btwn  <. A ,  C >. )  ->  P  Btwn  <. B ,  C >. ) )
2928adantr 463 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  A  Btwn  <. P ,  B >. ) )  -> 
( ( A  =/= 
P  /\  A  Btwn  <. B ,  P >.  /\  P  Btwn  <. A ,  C >. )  ->  P  Btwn  <. B ,  C >. ) )
3023, 25, 26, 29mp3and 1329 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  A  Btwn  <. P ,  B >. ) )  ->  P  Btwn  <. B ,  C >. )
3122, 30sylan2br 474 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  P  Btwn  <. A ,  C >. )  /\  A  Btwn  <. P ,  B >. ) )  ->  P  Btwn  <. B ,  C >. )
3231expr 613 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( A  Btwn  <. P ,  B >.  ->  P  Btwn  <. B ,  C >. ) )
33 df-3an 976 . . . . . . . 8  |-  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  B  Btwn  <. P ,  A >. )  <->  ( (
( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. )  /\  B  Btwn  <. P ,  A >. ) )
34 simpr3 1005 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  B  Btwn  <. P ,  A >. ) )  ->  B  Btwn  <. P ,  A >. )
355, 11, 6, 7, 34btwncomand 30340 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  B  Btwn  <. P ,  A >. ) )  ->  B  Btwn  <. A ,  P >. )
36 simpr2 1004 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  B  Btwn  <. P ,  A >. ) )  ->  P  Btwn  <. A ,  C >. )
375, 7, 11, 6, 8, 35, 36btwnexch3and 30346 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >.  /\  B  Btwn  <. P ,  A >. ) )  ->  P  Btwn  <. B ,  C >. )
3833, 37sylan2br 474 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/= 
P )  /\  P  Btwn  <. A ,  C >. )  /\  B  Btwn  <. P ,  A >. ) )  ->  P  Btwn  <. B ,  C >. )
3938expr 613 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( B  Btwn  <. P ,  A >.  ->  P  Btwn  <. B ,  C >. ) )
4032, 39jaod 378 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. )  ->  P  Btwn  <. B ,  C >. ) )
4121, 40syl5 30 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( ( A  =/=  P  /\  B  =/=  P  /\  ( A 
Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) )  ->  P  Btwn  <. B ,  C >. ) )
4220, 41impbid 191 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( P  Btwn  <. B ,  C >.  <-> 
( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
43 broutsideof2 30447 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( POutsideOf <. A ,  B >.  <-> 
( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
445, 6, 7, 11, 43syl13anc 1232 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( POutsideOf <. A ,  B >. 
<->  ( A  =/=  P  /\  B  =/=  P  /\  ( A  Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
4544adantr 463 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( POutsideOf <. A ,  B >.  <->  ( A  =/=  P  /\  B  =/=  P  /\  ( A 
Btwn  <. P ,  B >.  \/  B  Btwn  <. P ,  A >. ) ) ) )
4642, 45bitr4d 256 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
P  /\  B  =/=  P  /\  C  =/=  P
)  /\  P  Btwn  <. A ,  C >. ) )  ->  ( P  Btwn  <. B ,  C >.  <-> 
POutsideOf <. A ,  B >. ) )
4746ex 432 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  P  e.  ( EE `  N ) ) )  ->  ( ( ( A  =/=  P  /\  B  =/=  P  /\  C  =/=  P )  /\  P  Btwn  <. A ,  C >. )  ->  ( P  Btwn  <. B ,  C >.  <-> 
POutsideOf <. A ,  B >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    e. wcel 1842    =/= wne 2598   <.cop 3977   class class class wbr 4394   ` cfv 5568   NNcn 10575   EEcee 24595    Btwn cbtwn 24596  OutsideOfcoutsideof 30444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-ee 24598  df-btwn 24599  df-cgr 24600  df-ofs 30308  df-colinear 30364  df-ifs 30365  df-cgr3 30366  df-fs 30367  df-outsideof 30445
This theorem is referenced by: (None)
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