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Theorem isibg1a 25523
Description: An incidence-betweenness geometry is an incidence geometry. (For my private use only. Don't use.) (Contributed by FL, 2-Apr-2016.)
Hypothesis
Ref Expression
isibg1a.1  |-  ( ph  ->  G  e. Ibg )
Assertion
Ref Expression
isibg1a  |-  ( ph  ->  G  e. Ig )

Proof of Theorem isibg1a
Dummy variables  u  t  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isibg1a.1 . 2  |-  ( ph  ->  G  e. Ibg )
2 eqid 2283 . . . 4  |-  (PPoints `  G
)  =  (PPoints `  G
)
3 eqid 2283 . . . 4  |-  (PLines `  G )  =  (PLines `  G )
4 eqid 2283 . . . 4  |-  (btw `  G )  =  (btw
`  G )
5 eqid 2283 . . . 4  |-  (coln `  G )  =  (coln `  G )
62, 3, 4, 5isibg2 25522 . . 3  |-  ( G  e. Ibg 
<->  ( G  e. Ig  /\  A. z  e.  (PPoints `  G
) A. y  e.  (PPoints `  G )
( ( z  =/=  y  ->  E. w  e.  (PPoints `  G ) E. v  e.  (PPoints `  G ) E. u  e.  (PPoints `  G )
( z  e.  ( w (btw `  G
) y )  /\  v  e.  ( z
(btw `  G )
y )  /\  y  e.  ( z (btw `  G ) u ) ) )  /\  A. x  e.  (PPoints `  G
) ( ( ( { z ,  y ,  x }  e.  (coln `  G )  /\  ( z  =/=  y  /\  z  =/=  x  /\  y  =/=  x
) )  ->  (
( z  e.  ( y (btw `  G
) x )  /\  y  e/  ( z (btw
`  G ) x )  /\  x  e/  ( z (btw `  G ) y ) )  \/  ( z  e/  ( y (btw
`  G ) x )  /\  y  e.  ( z (btw `  G ) x )  /\  x  e/  (
z (btw `  G
) y ) )  \/  ( z  e/  ( y (btw `  G ) x )  /\  y  e/  (
z (btw `  G
) x )  /\  x  e.  ( z
(btw `  G )
y ) ) ) )  /\  ( ( y  e.  ( z (btw `  G )
x )  ->  (
y  e.  ( x (btw `  G )
z )  /\  {
z ,  y ,  x }  e.  (coln `  G )  /\  (
z  =/=  y  /\  z  =/=  x  /\  y  =/=  x ) ) )  /\  A. t  e.  (PLines `  G )
( ( z  e/  t  /\  y  e/  t  /\  x  e/  t
)  ->  ( (
( ( ( z (btw `  G )
y )  i^i  t
)  =  (/)  /\  (
( y (btw `  G ) x )  i^i  t )  =  (/) )  ->  ( ( z (btw `  G
) x )  i^i  t )  =  (/) )  /\  ( ( ( ( z (btw `  G ) y )  i^i  t )  =/=  (/)  /\  ( ( y (btw `  G )
x )  i^i  t
)  =/=  (/) )  -> 
( ( z (btw
`  G ) x )  i^i  t )  =  (/) ) ) ) ) ) ) ) )
76simplbi 446 . 2  |-  ( G  e. Ibg  ->  G  e. Ig )
81, 7syl 15 1  |-  ( ph  ->  G  e. Ig )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   A.wral 2543   E.wrex 2544    i^i cin 3151   (/)c0 3455   {ctp 3642   ` cfv 5255  (class class class)co 5858  PPointscpoints 25468  PLinescplines 25470  Igcig 25472  colnccol 25502  btwcbtw 25518  Ibgcibg 25519
This theorem is referenced by:  isibg1a6  25537  isibg1a7  25538  isibg1a8  25539  segline  25553  lppotos  25556  bsstrs  25558  rayline  25568  hpd  25581  abhp  25585  bhp3  25589
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-ibg2 25521
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