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Theorem btwnconn1 29314
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 4443 . . . . . 6  |-  ( B  =  C  ->  ( B  Btwn  <. A ,  D >.  <-> 
C  Btwn  <. A ,  D >. ) )
213anbi3d 1300 . . . . 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 1014 . . . . 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 997 . . . . . . 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 990 . . . . . . . 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 29313 . . . . . 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 2773 . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   <.cop 4026   class class class wbr 4440   ` cfv 5579   NNcn 10525   EEcee 23860    Btwn cbtwn 23861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-ee 23863  df-btwn 23864  df-cgr 23865  df-ofs 29196  df-colinear 29252  df-ifs 29253  df-cgr3 29254  df-fs 29255
This theorem is referenced by:  btwnconn2  29315  outsideoftr  29342  outsideofeq  29343  lineelsb2  29361
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