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Theorem hilbert1.2 26081
Description: There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.)
Assertion
Ref Expression
hilbert1.2  |-  ( P  =/=  Q  ->  E* x  e. LinesEE ( P  e.  x  /\  Q  e.  x ) )
Distinct variable groups:    x, P    x, Q

Proof of Theorem hilbert1.2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 an4 798 . . . . 5  |-  ( ( ( x  e. LinesEE  /\  y  e. LinesEE )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y ) ) )  <->  ( (
x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x ) )  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) ) )
2 simprl 733 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  x  e. LinesEE )
3 simprr 734 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  ( P  e.  x  /\  Q  e.  x )
)
4 simpl 444 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  P  =/=  Q )
5 linethru 26079 . . . . . . . . 9  |-  ( ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )  /\  P  =/=  Q
)  ->  x  =  ( PLine Q ) )
62, 3, 4, 5syl3anc 1184 . . . . . . . 8  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  x  =  ( PLine Q
) )
76ex 424 . . . . . . 7  |-  ( P  =/=  Q  ->  (
( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
)  ->  x  =  ( PLine Q ) ) )
8 simprl 733 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  y  e. LinesEE )
9 simprr 734 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  ( P  e.  y  /\  Q  e.  y )
)
10 simpl 444 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  P  =/=  Q )
11 linethru 26079 . . . . . . . . 9  |-  ( ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )  /\  P  =/=  Q
)  ->  y  =  ( PLine Q ) )
128, 9, 10, 11syl3anc 1184 . . . . . . . 8  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  y  =  ( PLine Q
) )
1312ex 424 . . . . . . 7  |-  ( P  =/=  Q  ->  (
( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
)  ->  y  =  ( PLine Q ) ) )
147, 13anim12d 547 . . . . . 6  |-  ( P  =/=  Q  ->  (
( ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x
) )  /\  (
y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y ) ) )  ->  ( x  =  ( PLine Q )  /\  y  =  ( PLine Q ) ) ) )
15 eqtr3 2454 . . . . . 6  |-  ( ( x  =  ( PLine Q )  /\  y  =  ( PLine Q
) )  ->  x  =  y )
1614, 15syl6 31 . . . . 5  |-  ( P  =/=  Q  ->  (
( ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x
) )  /\  (
y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y ) ) )  ->  x  =  y ) )
171, 16syl5bi 209 . . . 4  |-  ( P  =/=  Q  ->  (
( ( x  e. LinesEE  /\  y  e. LinesEE )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  x  =  y ) )
1817exp3a 426 . . 3  |-  ( P  =/=  Q  ->  (
( x  e. LinesEE  /\  y  e. LinesEE )  ->  ( (
( P  e.  x  /\  Q  e.  x
)  /\  ( P  e.  y  /\  Q  e.  y ) )  ->  x  =  y )
) )
1918ralrimivv 2789 . 2  |-  ( P  =/=  Q  ->  A. x  e. LinesEE 
A. y  e. LinesEE  (
( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
)  ->  x  =  y ) )
20 eleq2 2496 . . . 4  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
21 eleq2 2496 . . . 4  |-  ( x  =  y  ->  ( Q  e.  x  <->  Q  e.  y ) )
2220, 21anbi12d 692 . . 3  |-  ( x  =  y  ->  (
( P  e.  x  /\  Q  e.  x
)  <->  ( P  e.  y  /\  Q  e.  y ) ) )
2322rmo4 3119 . 2  |-  ( E* x  e. LinesEE ( P  e.  x  /\  Q  e.  x )  <->  A. x  e. LinesEE 
A. y  e. LinesEE  (
( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
)  ->  x  =  y ) )
2419, 23sylibr 204 1  |-  ( P  =/=  Q  ->  E* x  e. LinesEE ( P  e.  x  /\  Q  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E*wrmo 2700  (class class class)co 6073  Linecline2 26060  LinesEEclines2 26062
This theorem is referenced by:  linethrueu  26082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ec 6899  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ee 25822  df-btwn 25823  df-cgr 25824  df-ofs 25909  df-ifs 25965  df-cgr3 25966  df-colinear 25967  df-fs 25968  df-line2 26063  df-lines2 26065
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