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Theorem tglnssp 23764
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 23763 . 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 3585 . 2  |-  { z  e.  P  |  ( z  e.  ( X I Y )  \/  X  e.  ( z I Y )  \/  Y  e.  ( X I z ) ) }  C_  P
108, 9syl6eqss 3554 1  |-  ( ph  ->  ( X L Y )  C_  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 972    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818    C_ wss 3476   ` cfv 5588  (class class class)co 6285   Basecbs 14493  TarskiGcstrkg 23650  Itvcitv 23657  LineGclng 23658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-trkg 23675
This theorem is referenced by:  tglineelsb2  23823  tglinecom  23826
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