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Theorem btwnconn1 29719
Description: Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. (Contributed by Scott Fenton, 9-Oct-2013.)
Assertion
Ref Expression
btwnconn1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )

Proof of Theorem btwnconn1
StepHypRef Expression
1 breq1 4436 . . . . . 6  |-  ( B  =  C  ->  ( B  Btwn  <. A ,  D >.  <-> 
C  Btwn  <. A ,  D >. ) )
213anbi3d 1304 . . . . 5  |-  ( B  =  C  ->  (
( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. )  <-> 
( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. ) ) )
3 orc 385 . . . . . 6  |-  ( C 
Btwn  <. A ,  D >.  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
433ad2ant3 1018 . . . . 5  |-  ( ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
52, 4syl6bi 228 . . . 4  |-  ( B  =  C  ->  (
( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
65adantld 467 . . 3  |-  ( B  =  C  ->  (
( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
7 simpr1 1001 . . . . . . 7  |-  ( ( B  =/=  C  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  ->  A  =/=  B )
8 simpl 457 . . . . . . 7  |-  ( ( B  =/=  C  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  ->  B  =/=  C )
9 3simpc 994 . . . . . . . 8  |-  ( ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. )  ->  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )
109adantl 466 . . . . . . 7  |-  ( ( B  =/=  C  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  ->  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )
117, 8, 10jca31 534 . . . . . 6  |-  ( ( B  =/=  C  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  ->  ( ( A  =/=  B  /\  B  =/=  C )  /\  ( B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )
12 btwnconn1lem14 29718 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( ( A  =/= 
B  /\  B  =/=  C )  /\  ( B 
Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
1311, 12sylan2 474 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
1413an12s 799 . . . 4  |-  ( ( B  =/=  C  /\  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
1514ex 434 . . 3  |-  ( B  =/=  C  ->  (
( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
166, 15pm2.61ine 2754 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. ) )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) )
1716ex 434 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  B  Btwn  <. A ,  D >. )  ->  ( C  Btwn  <. A ,  D >.  \/  D  Btwn  <. A ,  C >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   <.cop 4016   class class class wbr 4433   ` cfv 5574   NNcn 10537   EEcee 24056    Btwn cbtwn 24057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483  df-ee 24059  df-btwn 24060  df-cgr 24061  df-ofs 29601  df-colinear 29657  df-ifs 29658  df-cgr3 29659  df-fs 29660
This theorem is referenced by:  btwnconn2  29720  outsideoftr  29747  outsideofeq  29748  lineelsb2  29766
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