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Theorem tglndim0 24210
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 729 . . . . 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 753 . . . . 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 24095 . . . 4  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  x  =  y )
7 simprr 755 . . . 4  |-  ( ( ( ( ( ph  /\  A  e.  ran  L
)  /\  x  e.  B )  /\  y  e.  B )  /\  ( A  =  ( x L y )  /\  x  =/=  y ) )  ->  x  =/=  y
)
86, 7pm2.21ddne 2768 . . 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 463 . . . 4  |-  ( (
ph  /\  A  e.  ran  L )  ->  G  e. TarskiG )
13 simpr 459 . . . 4  |-  ( (
ph  /\  A  e.  ran  L )  ->  A  e.  ran  L )
141, 9, 10, 12, 13tgisline 24208 . . 3  |-  ( (
ph  /\  A  e.  ran  L )  ->  E. x  e.  B  E. y  e.  B  ( A  =  ( x L y )  /\  x  =/=  y ) )
158, 14r19.29vva 2998 . 2  |-  ( (
ph  /\  A  e.  ran  L )  -> F.  )
1615inegd 1419 1  |-  ( ph  ->  -.  A  e.  ran  L )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398   F. wfal 1403    e. wcel 1823    =/= wne 2649   ran crn 4989   ` cfv 5570  (class class class)co 6270   1c1 9482   #chash 12387   Basecbs 14716  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388  df-trkg 24048
This theorem is referenced by:  hpgerlem  24335
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