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Theorem tglnssp 24496
Description: Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tglngval.p  |-  P  =  ( Base `  G
)
tglngval.l  |-  L  =  (LineG `  G )
tglngval.i  |-  I  =  (Itv `  G )
tglngval.g  |-  ( ph  ->  G  e. TarskiG )
tglngval.x  |-  ( ph  ->  X  e.  P )
tglngval.y  |-  ( ph  ->  Y  e.  P )
tglngval.z  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
tglnssp  |-  ( ph  ->  ( X L Y )  C_  P )

Proof of Theorem tglnssp
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tglngval.p . . 3  |-  P  =  ( Base `  G
)
2 tglngval.l . . 3  |-  L  =  (LineG `  G )
3 tglngval.i . . 3  |-  I  =  (Itv `  G )
4 tglngval.g . . 3  |-  ( ph  ->  G  e. TarskiG )
5 tglngval.x . . 3  |-  ( ph  ->  X  e.  P )
6 tglngval.y . . 3  |-  ( ph  ->  Y  e.  P )
7 tglngval.z . . 3  |-  ( ph  ->  X  =/=  Y )
81, 2, 3, 4, 5, 6, 7tglngval 24495 . 2  |-  ( ph  ->  ( X L Y )  =  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) } )
9 ssrab2 3543 . 2  |-  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) }  C_  P
108, 9syl6eqss 3511 1  |-  ( ph  ->  ( X L Y )  C_  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 981    = wceq 1437    e. wcel 1867    =/= wne 2616   {crab 2777    C_ wss 3433   ` cfv 5592  (class class class)co 6296   Basecbs 15081  TarskiGcstrkg 24380  Itvcitv 24386  LineGclng 24387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-trkg 24403
This theorem is referenced by:  tglineelsb2  24576  tglinecom  24579
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