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Theorem isibg1a7 26126
Description: If  Y is between  X and 
Z,  X belongs to the line YZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isibg1a3a.1  |-  P  =  (PPoints `  G )
isibg1a3a.2  |-  B  =  (btw `  G )
isibg1a3a.3  |-  ( ph  ->  G  e. Ibg )
isibg1a3a.4  |-  ( ph  ->  X  e.  P )
isibg1a3a.5  |-  ( ph  ->  Y  e.  P )
isibg1a3a.6  |-  ( ph  ->  Z  e.  P )
isibg1a3a.7  |-  ( ph  ->  Y  e.  ( X B Z ) )
isibg1a7.1  |-  M  =  ( line `  G
)
Assertion
Ref Expression
isibg1a7  |-  ( ph  ->  X  e.  ( Y M Z ) )

Proof of Theorem isibg1a7
StepHypRef Expression
1 isibg1a3a.1 . . 3  |-  P  =  (PPoints `  G )
2 isibg1a7.1 . . 3  |-  M  =  ( line `  G
)
3 isibg1a3a.3 . . . 4  |-  ( ph  ->  G  e. Ibg )
43isibg1a 26111 . . 3  |-  ( ph  ->  G  e. Ig )
5 isibg1a3a.4 . . 3  |-  ( ph  ->  X  e.  P )
6 isibg1a3a.6 . . 3  |-  ( ph  ->  Z  e.  P )
71, 2, 4, 5, 6lineval2a 26085 . 2  |-  ( ph  ->  X  e.  ( X M Z ) )
8 isibg1a3a.2 . . . 4  |-  B  =  (btw `  G )
9 isibg1a3a.5 . . . 4  |-  ( ph  ->  Y  e.  P )
10 isibg1a3a.7 . . . 4  |-  ( ph  ->  Y  e.  ( X B Z ) )
111, 8, 3, 5, 9, 6, 10, 2isibg1a6 26125 . . 3  |-  ( ph  ->  Y  e.  ( X M Z ) )
121, 8, 3, 5, 9, 6, 10isibg1a5a 26124 . . 3  |-  ( ph  ->  Y  =/=  Z )
131, 2, 4, 5, 6, 11, 12lineval5a 26088 . 2  |-  ( ph  ->  ( X M Z )  =  ( Y M Z ) )
147, 13eleqtrd 2359 1  |-  ( ph  ->  X  e.  ( Y M Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056   linecline 26076  btwcbtw 26106  Ibgcibg 26107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-ig2 26061  df-li 26077  df-col 26091  df-ibg2 26109
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