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Theorem btwnconn2 24065
Description: Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
Assertion
Ref Expression
btwnconn2  |-  ( ( 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  <. B ,  D >.  \/  D  Btwn  <. B ,  C >. ) ) )

Proof of Theorem btwnconn2
StepHypRef Expression
1 btwnconn1 24064 . 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 >. ) ) )
2 simpr2 967 . . . . . . 7  |-  ( ( ( 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 >. ) )  ->  B  Btwn  <. A ,  C >. )
32anim1i 554 . . . . . 6  |-  ( ( ( ( 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 >. )  ->  ( B  Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. ) )
4 btwnexch3 23983 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. )  ->  C  Btwn  <. B ,  D >. ) )
54ad2antrr 709 . . . . . 6  |-  ( ( ( ( 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 >. )  ->  ( ( B 
Btwn  <. A ,  C >.  /\  C  Btwn  <. A ,  D >. )  ->  C  Btwn  <. B ,  D >. ) )
63, 5mpd 16 . . . . 5  |-  ( ( ( ( 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 >. )  ->  C  Btwn  <. B ,  D >. )
76ex 425 . . . 4  |-  ( ( ( 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 >.  ->  C  Btwn  <. B ,  D >. ) )
8 simpr3 968 . . . . 5  |-  ( ( ( 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 >. ) )  ->  B  Btwn  <. A ,  D >. )
9 simp3r 989 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
10 simp3l 988 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
119, 10jca 520 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( D  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )
12 btwnexch3 23983 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  D >.  /\  D  Btwn  <. A ,  C >. )  ->  D  Btwn  <. B ,  C >. ) )
1311, 12syld3an3 1232 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  D >.  /\  D  Btwn  <. A ,  C >. )  ->  D  Btwn  <. B ,  C >. ) )
1413adantr 453 . . . . 5  |-  ( ( ( 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 >. ) )  ->  ( ( B  Btwn  <. A ,  D >.  /\  D  Btwn  <. A ,  C >. )  ->  D  Btwn  <. B ,  C >. ) )
158, 14mpand 659 . . . 4  |-  ( ( ( 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 >. ) )  ->  ( D  Btwn  <. A ,  C >.  ->  D  Btwn  <. B ,  C >. ) )
167, 15orim12d 814 . . 3  |-  ( ( ( 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 >. )  ->  ( C  Btwn  <. B ,  D >.  \/  D  Btwn  <. B ,  C >. ) ) )
1716ex 425 . 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 >. )  ->  ( C  Btwn  <. B ,  D >.  \/  D  Btwn  <. B ,  C >. ) ) ) )
181, 17mpdd 38 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  <. B ,  D >.  \/  D  Btwn  <. B ,  C >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    /\ w3a 939    e. wcel 1621    =/= wne 2419   <.cop 3584   class class class wbr 3963   ` cfv 4638   NNcn 9679   EEcee 23856    Btwn cbtwn 23857
This theorem is referenced by:  btwnconn3  24066  segcon2  24068  btwnoutside  24088  broutsideof3  24089  lineunray  24110  lineelsb2  24111
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-ee 23859  df-btwn 23860  df-cgr 23861  df-ofs 23946  df-ifs 24002  df-cgr3 24003  df-colinear 24004  df-fs 24005
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