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Theorem btwndiff 25873
Description: There is always a  c distinct from  B such that  B lies between  A and  c. Theorem 3.14 of [Schwabhauser] p. 32. (Contributed by Scott Fenton, 24-Sep-2013.)
Assertion
Ref Expression
btwndiff  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  E. c  e.  ( EE `  N
) ( B  Btwn  <. A ,  c >.  /\  B  =/=  c ) )
Distinct variable groups:    A, c    B, c    N, c

Proof of Theorem btwndiff
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axlowdim1 25810 . . 3  |-  ( N  e.  NN  ->  E. u  e.  ( EE `  N
) E. v  e.  ( EE `  N
) u  =/=  v
)
213ad2ant1 978 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  E. u  e.  ( EE `  N
) E. v  e.  ( EE `  N
) u  =/=  v
)
3 simp11 987 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  ->  N  e.  NN )
4 simp12 988 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  ->  A  e.  ( EE `  N ) )
5 simp13 989 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  ->  B  e.  ( EE `  N ) )
6 simp2l 983 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  ->  u  e.  ( EE `  N ) )
7 simp2r 984 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  -> 
v  e.  ( EE
`  N ) )
8 axsegcon 25778 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) ) )  ->  E. c  e.  ( EE `  N ) ( B  Btwn  <. A , 
c >.  /\  <. B , 
c >.Cgr <. u ,  v
>. ) )
93, 4, 5, 6, 7, 8syl122anc 1193 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  ->  E. c  e.  ( EE `  N ) ( B  Btwn  <. A , 
c >.  /\  <. B , 
c >.Cgr <. u ,  v
>. ) )
10 simpl11 1032 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  N  e.  NN )
11 simpl13 1034 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
12 simpr 448 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  c  e.  ( EE `  N
) )
13 simpl2l 1010 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  u  e.  ( EE `  N
) )
14 simpl2r 1011 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  v  e.  ( EE `  N
) )
15 cgrdegen 25850 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  c  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) ) )  ->  ( <. B , 
c >.Cgr <. u ,  v
>.  ->  ( B  =  c  <->  u  =  v
) ) )
1610, 11, 12, 13, 14, 15syl122anc 1193 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  ( <. B ,  c >.Cgr <. u ,  v >.  ->  ( B  =  c  <-> 
u  =  v ) ) )
17 bi1 179 . . . . . . . . . . . 12  |-  ( ( B  =  c  <->  u  =  v )  ->  ( B  =  c  ->  u  =  v ) )
1817necon3d 2613 . . . . . . . . . . 11  |-  ( ( B  =  c  <->  u  =  v )  ->  (
u  =/=  v  ->  B  =/=  c ) )
1918com12 29 . . . . . . . . . 10  |-  ( u  =/=  v  ->  (
( B  =  c  <-> 
u  =  v )  ->  B  =/=  c
) )
20193ad2ant3 980 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  -> 
( ( B  =  c  <->  u  =  v
)  ->  B  =/=  c ) )
2120adantr 452 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  (
( B  =  c  <-> 
u  =  v )  ->  B  =/=  c
) )
2216, 21syld 42 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  ( <. B ,  c >.Cgr <. u ,  v >.  ->  B  =/=  c ) )
2322anim2d 549 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N
) )  /\  u  =/=  v )  /\  c  e.  ( EE `  N
) )  ->  (
( B  Btwn  <. A , 
c >.  /\  <. B , 
c >.Cgr <. u ,  v
>. )  ->  ( B 
Btwn  <. A ,  c
>.  /\  B  =/=  c
) ) )
2423reximdva 2786 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  -> 
( E. c  e.  ( EE `  N
) ( B  Btwn  <. A ,  c >.  /\ 
<. B ,  c >.Cgr <. u ,  v >.
)  ->  E. c  e.  ( EE `  N
) ( B  Btwn  <. A ,  c >.  /\  B  =/=  c ) ) )
259, 24mpd 15 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  /\  u  =/=  v )  ->  E. c  e.  ( EE `  N ) ( B  Btwn  <. A , 
c >.  /\  B  =/=  c ) )
26253exp 1152 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  (
( u  e.  ( EE `  N )  /\  v  e.  ( EE `  N ) )  ->  ( u  =/=  v  ->  E. c  e.  ( EE `  N
) ( B  Btwn  <. A ,  c >.  /\  B  =/=  c ) ) ) )
2726rexlimdvv 2804 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  ( E. u  e.  ( EE `  N ) E. v  e.  ( EE
`  N ) u  =/=  v  ->  E. c  e.  ( EE `  N
) ( B  Btwn  <. A ,  c >.  /\  B  =/=  c ) ) )
282, 27mpd 15 1  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  E. c  e.  ( EE `  N
) ( B  Btwn  <. A ,  c >.  /\  B  =/=  c ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   <.cop 3785   class class class wbr 4180   ` cfv 5421   NNcn 9964   EEcee 25739    Btwn cbtwn 25740  Cgrccgr 25741
This theorem is referenced by:  ifscgr  25890  cgrxfr  25901  btwnconn3  25949  broutsideof3  25972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-ee 25742  df-btwn 25743  df-cgr 25744
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