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Theorem hilbert1.2 25804
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 25802 . . . . . . . . 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 25802 . . . . . . . . 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 2407 . . . . . 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 2741 . 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 2449 . . . 4  |-  ( x  =  y  ->  ( P  e.  x  <->  P  e.  y ) )
21 eleq2 2449 . . . 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 3071 . 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 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E*wrmo 2653  (class class class)co 6021  Linecline2 25783  LinesEEclines2 25785
This theorem is referenced by:  linethrueu  25805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-ec 6844  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-sum 12408  df-ee 25545  df-btwn 25546  df-cgr 25547  df-ofs 25632  df-ifs 25688  df-cgr3 25689  df-colinear 25690  df-fs 25691  df-line2 25786  df-lines2 25788
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