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Theorem hilbert1.1 29731
Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
hilbert1.1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  E. x  e. LinesEE  ( P  e.  x  /\  Q  e.  x ) )
Distinct variable groups:    x, P    x, Q
Allowed substitution hint:    N( x)

Proof of Theorem hilbert1.1
Dummy variables  n  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . . . 5  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  ->  P  e.  ( EE `  N
) )
2 simp2 997 . . . . 5  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  ->  Q  e.  ( EE `  N
) )
3 simp3 998 . . . . 5  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  ->  P  =/=  Q )
4 eqidd 2468 . . . . 5  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  ->  ( PLine Q )  =  ( PLine Q ) )
5 neeq1 2748 . . . . . . 7  |-  ( p  =  P  ->  (
p  =/=  q  <->  P  =/=  q ) )
6 oveq1 6302 . . . . . . . 8  |-  ( p  =  P  ->  (
pLine q )  =  ( PLine q ) )
76eqeq2d 2481 . . . . . . 7  |-  ( p  =  P  ->  (
( PLine Q )  =  ( pLine q
)  <->  ( PLine Q
)  =  ( PLine q ) ) )
85, 7anbi12d 710 . . . . . 6  |-  ( p  =  P  ->  (
( p  =/=  q  /\  ( PLine Q )  =  ( pLine q
) )  <->  ( P  =/=  q  /\  ( PLine Q )  =  ( PLine q ) ) ) )
9 neeq2 2750 . . . . . . 7  |-  ( q  =  Q  ->  ( P  =/=  q  <->  P  =/=  Q ) )
10 oveq2 6303 . . . . . . . 8  |-  ( q  =  Q  ->  ( PLine q )  =  ( PLine Q ) )
1110eqeq2d 2481 . . . . . . 7  |-  ( q  =  Q  ->  (
( PLine Q )  =  ( PLine q
)  <->  ( PLine Q
)  =  ( PLine Q ) ) )
129, 11anbi12d 710 . . . . . 6  |-  ( q  =  Q  ->  (
( P  =/=  q  /\  ( PLine Q )  =  ( PLine q
) )  <->  ( P  =/=  Q  /\  ( PLine Q )  =  ( PLine Q ) ) ) )
138, 12rspc2ev 3230 . . . . 5  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  ( P  =/=  Q  /\  ( PLine Q )  =  ( PLine Q ) ) )  ->  E. p  e.  ( EE `  N
) E. q  e.  ( EE `  N
) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) ) )
141, 2, 3, 4, 13syl112anc 1232 . . . 4  |-  ( ( P  e.  ( EE
`  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q )  ->  E. p  e.  ( EE `  N
) E. q  e.  ( EE `  N
) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) ) )
15 fveq2 5872 . . . . . 6  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
1615rexeqdv 3070 . . . . . 6  |-  ( n  =  N  ->  ( E. q  e.  ( EE `  n ) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) )  <->  E. q  e.  ( EE `  N ) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q
) ) ) )
1715, 16rexeqbidv 3078 . . . . 5  |-  ( n  =  N  ->  ( E. p  e.  ( EE `  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) )  <->  E. p  e.  ( EE `  N ) E. q  e.  ( EE `  N ) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q
) ) ) )
1817rspcev 3219 . . . 4  |-  ( ( N  e.  NN  /\  E. p  e.  ( EE
`  N ) E. q  e.  ( EE
`  N ) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) ) )  ->  E. n  e.  NN  E. p  e.  ( EE `  n
) E. q  e.  ( EE `  n
) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) ) )
1914, 18sylan2 474 . . 3  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) ) )
20 ellines 29729 . . 3  |-  ( ( PLine Q )  e. LinesEE  <->  E. n  e.  NN  E. p  e.  ( EE `  n ) E. q  e.  ( EE `  n
) ( p  =/=  q  /\  ( PLine Q )  =  ( pLine q ) ) )
2119, 20sylibr 212 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  -> 
( PLine Q )  e. LinesEE )
22 linerflx1 29726 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  P  e.  ( PLine Q ) )
23 linerflx2 29728 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  Q  e.  ( PLine Q ) )
24 eleq2 2540 . . . 4  |-  ( x  =  ( PLine Q
)  ->  ( P  e.  x  <->  P  e.  ( PLine Q ) ) )
25 eleq2 2540 . . . 4  |-  ( x  =  ( PLine Q
)  ->  ( Q  e.  x  <->  Q  e.  ( PLine Q ) ) )
2624, 25anbi12d 710 . . 3  |-  ( x  =  ( PLine Q
)  ->  ( ( P  e.  x  /\  Q  e.  x )  <->  ( P  e.  ( PLine Q )  /\  Q  e.  ( PLine Q ) ) ) )
2726rspcev 3219 . 2  |-  ( ( ( PLine Q )  e. LinesEE  /\  ( P  e.  ( PLine Q )  /\  Q  e.  ( PLine Q ) ) )  ->  E. x  e. LinesEE  ( P  e.  x  /\  Q  e.  x
) )
2821, 22, 23, 27syl12anc 1226 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  Q  e.  ( EE `  N )  /\  P  =/=  Q ) )  ->  E. x  e. LinesEE  ( P  e.  x  /\  Q  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   ` cfv 5594  (class class class)co 6295   NNcn 10548   EEcee 24014  Linecline2 29711  LinesEEclines2 29713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-ec 7325  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-ee 24017  df-btwn 24018  df-cgr 24019  df-colinear 29616  df-line2 29714  df-lines2 29716
This theorem is referenced by:  linethrueu  29733
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