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Theorem tglndim0 23169
Description: There are no lines in dimension 0. (Contributed by Thierry Arnoux, 18-Oct-2019.)
Hypotheses
Ref Expression
tglineelsb2.p  |-  B  =  ( Base `  G
)
tglineelsb2.i  |-  I  =  (Itv `  G )
tglineelsb2.l  |-  L  =  (LineG `  G )
tglineelsb2.g  |-  ( ph  ->  G  e. TarskiG )
tglndim0.d  |-  ( ph  ->  ( # `  B
)  =  1 )
Assertion
Ref Expression
tglndim0  |-  ( ph  ->  -.  A  e.  ran  L )

Proof of Theorem tglndim0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . . . 5  |-  B  =  ( Base `  G
)
2 tglndim0.d . . . . . 6  |-  ( ph  ->  ( # `  B
)  =  1 )
32ad4antr 731 . . . . 5  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  ( # `  B
)  =  1 )
4 simpllr 758 . . . . 5  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  x  e.  B
)
5 simplr 754 . . . . 5  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  y  e.  B
)
61, 3, 4, 5tgldim0eq 23086 . . . 4  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  x  =  y )
7 simprr 756 . . . 4  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  x  =/=  y
)
86, 7pm2.21ddne 2763 . . 3  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  -> F.  )
9 tglineelsb2.i . . . 4  |-  I  =  (Itv `  G )
10 tglineelsb2.l . . . 4  |-  L  =  (LineG `  G )
11 tglineelsb2.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
1211adantr 465 . . . 4  |-  ( (
ph  /\  A  e.  ran  L )  ->  G  e. TarskiG )
13 simpr 461 . . . 4  |-  ( (
ph  /\  A  e.  ran  L )  ->  A  e.  ran  L )
141, 9, 10, 12, 13tgisline 23167 . . 3  |-  ( (
ph  /\  A  e.  ran  L )  ->  E. x  e.  B  E. y  e.  B  ( A  =  ( x L y )  /\  x  =/=  y ) )
158, 14r19.29_2a 2964 . 2  |-  ( (
ph  /\  A  e.  ran  L )  -> F.  )
1615inegd 1391 1  |-  ( ph  ->  -.  A  e.  ran  L )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   F. wfal 1375    e. wcel 1758    =/= wne 2645   ran crn 4944   ` cfv 5521  (class class class)co 6195   1c1 9389   #chash 12215   Basecbs 14287  TarskiGcstrkg 23017  Itvcitv 23024  LineGclng 23025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-hash 12216  df-trkg 23042
This theorem is referenced by: (None)
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