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Theorem btwnouttr2 30782
Description: Outer transitivity law for betweenness. Left-hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
btwnouttr2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )

Proof of Theorem btwnouttr2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp1 1005 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  N  e.  NN )
2 simp2l 1031 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
3 simp3l 1033 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
4 simp3r 1034 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
5 axsegcon 24944 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. ) )
61, 2, 3, 3, 4, 5syl122anc 1273 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. ) )
76adantr 466 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  E. x  e.  ( EE `  N
) ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )
8 simprrl 772 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. A ,  x >. )
9 simprl1 1050 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  B  =/=  C )
10 simpl2 1009 . . . . . . . . . . . . 13  |-  ( ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )  ->  B  Btwn  <. A ,  C >. )
11 simprl 762 . . . . . . . . . . . . 13  |-  ( ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )  ->  C  Btwn  <. A ,  x >. )
1210, 11jca 534 . . . . . . . . . . . 12  |-  ( ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\ 
<. C ,  x >.Cgr <. C ,  D >. ) )  ->  ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. ) )
1312adantl 467 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. ) )
14 simpl1 1008 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  N  e.  NN )
15 simpl2l 1058 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
16 simpl2r 1059 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  B  e.  ( EE `  N ) )
17 simpl3l 1060 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  C  e.  ( EE `  N ) )
18 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  x  e.  ( EE `  N ) )
19 btwnexch3 30780 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. )  ->  C  Btwn  <. B ,  x >. ) )
2014, 15, 16, 17, 18, 19syl122anc 1273 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. )  ->  C  Btwn  <. B ,  x >. ) )
2120adantr 466 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  x >. )  ->  C  Btwn  <. B ,  x >. ) )
2213, 21mpd 15 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. B ,  x >. )
23 simprrr 773 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  <. C ,  x >.Cgr <. C ,  D >. )
2422, 23jca 534 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) )
25 simprl3 1052 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. B ,  D >. )
26 simpl3r 1061 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  D  e.  ( EE `  N ) )
2714, 17, 26cgrrflxd 30748 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  <. C ,  D >.Cgr <. C ,  D >. )
2827adantr 466 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  <. C ,  D >.Cgr <. C ,  D >. )
2925, 28jca 534 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  ( C  Btwn  <. B ,  D >.  /\  <. C ,  D >.Cgr
<. C ,  D >. ) )
30 segconeq 30770 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  x  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( B  =/=  C  /\  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. B ,  D >.  /\  <. C ,  D >.Cgr
<. C ,  D >. ) )  ->  x  =  D ) )
3114, 17, 17, 26, 16, 18, 26, 30syl133anc 1287 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( B  =/= 
C  /\  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. )  /\  ( C  Btwn  <. B ,  D >.  /\ 
<. C ,  D >.Cgr <. C ,  D >. ) )  ->  x  =  D ) )
3231adantr 466 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  (
( B  =/=  C  /\  ( C  Btwn  <. B ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. )  /\  ( C 
Btwn  <. B ,  D >.  /\  <. C ,  D >.Cgr
<. C ,  D >. ) )  ->  x  =  D ) )
339, 24, 29, 32mp3and 1363 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  x  =  D )
3433opeq2d 4191 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  <. A ,  x >.  =  <. A ,  D >. )
358, 34breqtrd 4445 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( B  =/= 
C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  /\  ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. ) ) )  ->  C  Btwn  <. A ,  D >. )
3635expr 618 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  ( ( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
3736an32s 811 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  /\  x  e.  ( EE `  N
) )  ->  (
( C  Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr <. C ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
3837rexlimdva 2917 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  ( E. x  e.  ( EE `  N ) ( C 
Btwn  <. A ,  x >.  /\  <. C ,  x >.Cgr
<. C ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
397, 38mpd 15 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  C  Btwn  <. A ,  D >. )
4039ex 435 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  ->  C  Btwn  <. A ,  D >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776   <.cop 4002   class class class wbr 4420   ` cfv 5598   NNcn 10610   EEcee 24905    Btwn cbtwn 24906  Cgrccgr 24907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-ee 24908  df-btwn 24909  df-cgr 24910  df-ofs 30743
This theorem is referenced by:  btwnexch2  30783  btwnouttr  30784  btwnoutside  30885  lineelsb2  30908
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