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Definition df-seg2 25543
Description: Definition of the segment xy degenerated or not. Definition 8 of [AitkenIBG] p. 4. (For my private use only. Don't use.) (Contributed by FL, 28-Feb-2016.)
Assertion
Ref Expression
df-seg2  |-  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
Distinct variable group:    x, f, y, z

Detailed syntax breakdown of Definition df-seg2
StepHypRef Expression
1 cseg 25542 . 2  class  seg
2 vf . . 3  set  f
3 cibg 25519 . . 3  class Ibg
4 vx . . . 4  set  x
5 vy . . . 4  set  y
62cv 1622 . . . . 5  class  f
7 cpoints 25468 . . . . 5  class PPoints
86, 7cfv 5255 . . . 4  class  (PPoints `  f
)
94cv 1622 . . . . . 6  class  x
105cv 1622 . . . . . 6  class  y
119, 10wne 2446 . . . . 5  wff  x  =/=  y
12 vz . . . . . . . . 9  set  z
1312cv 1622 . . . . . . . 8  class  z
14 cbtw 25518 . . . . . . . . . 10  class btw
156, 14cfv 5255 . . . . . . . . 9  class  (btw `  f )
169, 10, 15co 5858 . . . . . . . 8  class  ( x (btw `  f )
y )
1713, 16wcel 1684 . . . . . . 7  wff  z  e.  ( x (btw `  f ) y )
1812, 4weq 1624 . . . . . . 7  wff  z  =  x
1912, 5weq 1624 . . . . . . 7  wff  z  =  y
2017, 18, 19w3o 933 . . . . . 6  wff  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y )
2120, 12, 8crab 2547 . . . . 5  class  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) }
229csn 3640 . . . . 5  class  { x }
2311, 21, 22cif 3565 . . . 4  class  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
)
244, 5, 8, 8, 23cmpt2 5860 . . 3  class  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )
252, 3, 24cmpt 4077 . 2  class  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
) ) )
261, 25wceq 1623 1  wff  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
Colors of variables: wff set class
This definition is referenced by:  sgplpte21  25544  sgplpte22  25550
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