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Theorem tgbtwnconn2 23008
Description: Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tgbtwnconn.p  |-  P  =  ( Base `  G
)
tgbtwnconn.i  |-  I  =  (Itv `  G )
tgbtwnconn.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwnconn.a  |-  ( ph  ->  A  e.  P )
tgbtwnconn.b  |-  ( ph  ->  B  e.  P )
tgbtwnconn.c  |-  ( ph  ->  C  e.  P )
tgbtwnconn.d  |-  ( ph  ->  D  e.  P )
tgbtwnconn2.1  |-  ( ph  ->  A  =/=  B )
tgbtwnconn2.2  |-  ( ph  ->  B  e.  ( A I C ) )
tgbtwnconn2.3  |-  ( ph  ->  B  e.  ( A I D ) )
Assertion
Ref Expression
tgbtwnconn2  |-  ( ph  ->  ( C  e.  ( B I D )  \/  D  e.  ( B I C ) ) )

Proof of Theorem tgbtwnconn2
StepHypRef Expression
1 tgbtwnconn.p . . . 4  |-  P  =  ( Base `  G
)
2 eqid 2443 . . . 4  |-  ( dist `  G )  =  (
dist `  G )
3 tgbtwnconn.i . . . 4  |-  I  =  (Itv `  G )
4 tgbtwnconn.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  G  e. TarskiG )
6 tgbtwnconn.a . . . . 5  |-  ( ph  ->  A  e.  P )
76adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  A  e.  P )
8 tgbtwnconn.b . . . . 5  |-  ( ph  ->  B  e.  P )
98adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  B  e.  P )
10 tgbtwnconn.c . . . . 5  |-  ( ph  ->  C  e.  P )
1110adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  C  e.  P )
12 tgbtwnconn.d . . . . 5  |-  ( ph  ->  D  e.  P )
1312adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  D  e.  P )
14 tgbtwnconn2.2 . . . . 5  |-  ( ph  ->  B  e.  ( A I C ) )
1514adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  B  e.  ( A I C ) )
16 simpr 461 . . . 4  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  C  e.  ( A I D ) )
171, 2, 3, 5, 7, 9, 11, 13, 15, 16tgbtwnexch3 22947 . . 3  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  C  e.  ( B I D ) )
1817orcd 392 . 2  |-  ( (
ph  /\  C  e.  ( A I D ) )  ->  ( C  e.  ( B I D )  \/  D  e.  ( B I C ) ) )
194adantr 465 . . . 4  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  G  e. TarskiG )
206adantr 465 . . . 4  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  A  e.  P )
218adantr 465 . . . 4  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  B  e.  P )
2212adantr 465 . . . 4  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  D  e.  P )
2310adantr 465 . . . 4  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  C  e.  P )
24 tgbtwnconn2.3 . . . . 5  |-  ( ph  ->  B  e.  ( A I D ) )
2524adantr 465 . . . 4  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  B  e.  ( A I D ) )
26 simpr 461 . . . 4  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  D  e.  ( A I C ) )
271, 2, 3, 19, 20, 21, 22, 23, 25, 26tgbtwnexch3 22947 . . 3  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  D  e.  ( B I C ) )
2827olcd 393 . 2  |-  ( (
ph  /\  D  e.  ( A I C ) )  ->  ( C  e.  ( B I D )  \/  D  e.  ( B I C ) ) )
29 tgbtwnconn2.1 . . 3  |-  ( ph  ->  A  =/=  B )
301, 3, 4, 6, 8, 10, 12, 29, 14, 24tgbtwnconn1 23007 . 2  |-  ( ph  ->  ( C  e.  ( A I D )  \/  D  e.  ( A I C ) ) )
3118, 28, 30mpjaodan 784 1  |-  ( ph  ->  ( C  e.  ( B I D )  \/  D  e.  ( B I C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   ` cfv 5418  (class class class)co 6091   Basecbs 14174   distcds 14247  TarskiGcstrkg 22889  Itvcitv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-s2 12475  df-s3 12476  df-trkgc 22909  df-trkgb 22910  df-trkgcb 22911  df-trkg 22916  df-cgrg 22964
This theorem is referenced by:  tgbtwnconn3  23009  tgbtwnconnln2  23012  legtrid  23022  krippenlem  23084
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