Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  btwndiff Unicode version

Theorem btwndiff 25677
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 25614 . . 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 25582 . . . . . 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 25654 . . . . . . . . 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 2590 . . . . . . . . . . 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 2763 . . . . 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 2781 . 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 1717    =/= wne 2552   E.wrex 2652   <.cop 3762   class class class wbr 4155   ` cfv 5396   NNcn 9934   EEcee 25543    Btwn cbtwn 25544  Cgrccgr 25545
This theorem is referenced by:  ifscgr  25694  cgrxfr  25705  btwnconn3  25753  broutsideof3  25776
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-sum 12409  df-ee 25546  df-btwn 25547  df-cgr 25548
  Copyright terms: Public domain W3C validator