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Theorem btwnouttr2 28198
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 988 . . . . 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 1014 . . . . 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 1016 . . . . 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 1017 . . . . 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 23326 . . . . 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 1228 . . . 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 465 . . 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 763 . . . . . . 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 1033 . . . . . . . . 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 992 . . . . . . . . . . . . 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 755 . . . . . . . . . . . . 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 532 . . . . . . . . . . . 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 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 >. ) )
14 simpl1 991 . . . . . . . . . . . . 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 1041 . . . . . . . . . . . . 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 1042 . . . . . . . . . . . . 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 1043 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 28196 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 764 . . . . . . . . . 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 532 . . . . . . . . 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 1035 . . . . . . . . . 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 1044 . . . . . . . . . . . 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 28164 . . . . . . . . . . 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 465 . . . . . . . . . 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 532 . . . . . . . . 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 28186 . . . . . . . . . . 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 1242 . . . . . . . . . 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 465 . . . . . . . . 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 1318 . . . . . . . 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 4175 . . . . . . 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 4425 . . . . . 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 615 . . . . 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 802 . . . 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 2947 . . 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 434 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   <.cop 3992   class class class wbr 4401   ` cfv 5527   NNcn 10434   EEcee 23287    Btwn cbtwn 23288  Cgrccgr 23289
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-ico 11418  df-icc 11419  df-fz 11556  df-fzo 11667  df-seq 11925  df-exp 11984  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-sum 13283  df-ee 23290  df-btwn 23291  df-cgr 23292  df-ofs 28159
This theorem is referenced by:  btwnexch2  28199  btwnouttr  28200  btwnoutside  28301  lineelsb2  28324
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