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Theorem isibg1a 26214
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 2296 . . . 4  |-  (PPoints `  G
)  =  (PPoints `  G
)
3 eqid 2296 . . . 4  |-  (PLines `  G )  =  (PLines `  G )
4 eqid 2296 . . . 4  |-  (btw `  G )  =  (btw
`  G )
5 eqid 2296 . . . 4  |-  (coln `  G )  =  (coln `  G )
62, 3, 4, 5isibg2 26213 . . 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 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460   A.wral 2556   E.wrex 2557    i^i cin 3164   (/)c0 3468   {ctp 3655   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Igcig 26163  colnccol 26193  btwcbtw 26209  Ibgcibg 26210
This theorem is referenced by:  isibg1a6  26228  isibg1a7  26229  isibg1a8  26230  segline  26244  lppotos  26247  bsstrs  26249  rayline  26259  hpd  26272  abhp  26276  bhp3  26280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-ibg2 26212
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