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Theorem btwndiff 30799
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 24987 . . 3  |-  ( N  e.  NN  ->  E. u  e.  ( EE `  N
) E. v  e.  ( EE `  N
) u  =/=  v
)
213ad2ant1 1026 . 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 1035 . . . . . 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 1036 . . . . . 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 1037 . . . . . 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 1031 . . . . . 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 1032 . . . . . 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 24955 . . . . . 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 1273 . . . . 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 1080 . . . . . . . . 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 1082 . . . . . . . . 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 462 . . . . . . . . 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 1058 . . . . . . . . 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 1059 . . . . . . . . 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 30776 . . . . . . . . 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 1273 . . . . . . . 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 biimp 196 . . . . . . . . . . . 12  |-  ( ( B  =  c  <->  u  =  v )  ->  ( B  =  c  ->  u  =  v ) )
1817necon3d 2644 . . . . . . . . . . 11  |-  ( ( B  =  c  <->  u  =  v )  ->  (
u  =/=  v  ->  B  =/=  c ) )
1918com12 32 . . . . . . . . . 10  |-  ( u  =/=  v  ->  (
( B  =  c  <-> 
u  =  v )  ->  B  =/=  c
) )
20193ad2ant3 1028 . . . . . . . . 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 466 . . . . . . . 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 45 . . . . . . 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 567 . . . . . 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 2897 . . . . 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 1204 . . 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 2920 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   <.cop 4004   class class class wbr 4423   ` cfv 5601   NNcn 10616   EEcee 24916    Btwn cbtwn 24917  Cgrccgr 24918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  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 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-sum 13752  df-ee 24919  df-btwn 24920  df-cgr 24921
This theorem is referenced by:  ifscgr  30816  cgrxfr  30827  btwnconn3  30875  broutsideof3  30898
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