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Theorem tghilberti2 24681
Description: There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Thierry Arnoux, 25-May-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 )
tglineelsb2.1  |-  ( ph  ->  P  e.  B )
tglineelsb2.2  |-  ( ph  ->  Q  e.  B )
tglineelsb2.4  |-  ( ph  ->  P  =/=  Q )
Assertion
Ref Expression
tghilberti2  |-  ( ph  ->  E* x  e.  ran  L ( P  e.  x  /\  Q  e.  x
) )
Distinct variable groups:    x, B    x, G    x, I    x, L    x, P    x, Q    ph, x

Proof of Theorem tghilberti2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 tglineelsb2.p . . . . . 6  |-  B  =  ( Base `  G
)
2 tglineelsb2.i . . . . . 6  |-  I  =  (Itv `  G )
3 tglineelsb2.l . . . . . 6  |-  L  =  (LineG `  G )
4 tglineelsb2.g . . . . . . 7  |-  ( ph  ->  G  e. TarskiG )
543ad2ant1 1026 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  G  e. TarskiG )
6 tglineelsb2.1 . . . . . . 7  |-  ( ph  ->  P  e.  B )
763ad2ant1 1026 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  P  e.  B )
8 tglineelsb2.2 . . . . . . 7  |-  ( ph  ->  Q  e.  B )
983ad2ant1 1026 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  Q  e.  B )
10 tglineelsb2.4 . . . . . . 7  |-  ( ph  ->  P  =/=  Q )
11103ad2ant1 1026 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  P  =/=  Q )
12 simp2l 1031 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  x  e.  ran  L )
13 simp3ll 1076 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  P  e.  x )
14 simp3lr 1077 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  Q  e.  x )
151, 2, 3, 5, 7, 9, 11, 11, 12, 13, 14tglinethru 24679 . . . . 5  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  x  =  ( P L Q ) )
16 simp2r 1032 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  y  e.  ran  L )
17 simp3rl 1078 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  P  e.  y )
18 simp3rr 1079 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  Q  e.  y )
191, 2, 3, 5, 7, 9, 11, 11, 16, 17, 18tglinethru 24679 . . . . 5  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  y  =  ( P L Q ) )
2015, 19eqtr4d 2466 . . . 4  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  x  =  y )
21203expia 1207 . . 3  |-  ( (
ph  /\  ( x  e.  ran  L  /\  y  e.  ran  L ) )  ->  ( ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y ) )  ->  x  =  y ) )
2221ralrimivva 2843 . 2  |-  ( ph  ->  A. x  e.  ran  L A. y  e.  ran  L ( ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
)  ->  x  =  y ) )
23 eleq2 2496 . . . 4  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
24 eleq2 2496 . . . 4  |-  ( x  =  y  ->  ( Q  e.  x  <->  Q  e.  y ) )
2523, 24anbi12d 715 . . 3  |-  ( x  =  y  ->  (
( P  e.  x  /\  Q  e.  x
)  <->  ( P  e.  y  /\  Q  e.  y ) ) )
2625rmo4 3263 . 2  |-  ( E* x  e.  ran  L
( P  e.  x  /\  Q  e.  x
)  <->  A. x  e.  ran  L A. y  e.  ran  L ( ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
)  ->  x  =  y ) )
2722, 26sylibr 215 1  |-  ( ph  ->  E* x  e.  ran  L ( P  e.  x  /\  Q  e.  x
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E*wrmo 2774   ran crn 4854   ` cfv 5601  (class class class)co 6305   Basecbs 15120  TarskiGcstrkg 24476  Itvcitv 24482  LineGclng 24483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-concat 12670  df-s1 12671  df-s2 12946  df-s3 12947  df-trkgc 24494  df-trkgb 24495  df-trkgcb 24496  df-trkg 24499  df-cgrg 24554
This theorem is referenced by:  tglinethrueu  24682
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