Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hilbert1.2 Structured version   Unicode version

Theorem hilbert1.2 30927
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 831 . . . . 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 762 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  x  e. LinesEE )
3 simprr 764 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  ( P  e.  x  /\  Q  e.  x )
)
4 simpl 458 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  P  =/=  Q )
5 linethru 30925 . . . . . . . . 9  |-  ( ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )  /\  P  =/=  Q
)  ->  x  =  ( PLine Q ) )
62, 3, 4, 5syl3anc 1264 . . . . . . . 8  |-  ( ( P  =/=  Q  /\  ( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
) )  ->  x  =  ( PLine Q
) )
76ex 435 . . . . . . 7  |-  ( P  =/=  Q  ->  (
( x  e. LinesEE  /\  ( P  e.  x  /\  Q  e.  x )
)  ->  x  =  ( PLine Q ) ) )
8 simprl 762 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  y  e. LinesEE )
9 simprr 764 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  ( P  e.  y  /\  Q  e.  y )
)
10 simpl 458 . . . . . . . . 9  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  P  =/=  Q )
11 linethru 30925 . . . . . . . . 9  |-  ( ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )  /\  P  =/=  Q
)  ->  y  =  ( PLine Q ) )
128, 9, 10, 11syl3anc 1264 . . . . . . . 8  |-  ( ( P  =/=  Q  /\  ( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  y  =  ( PLine Q
) )
1312ex 435 . . . . . . 7  |-  ( P  =/=  Q  ->  (
( y  e. LinesEE  /\  ( P  e.  y  /\  Q  e.  y )
)  ->  y  =  ( PLine Q ) ) )
147, 13anim12d 565 . . . . . 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 2450 . . . . . 6  |-  ( ( x  =  ( PLine Q )  /\  y  =  ( PLine Q
) )  ->  x  =  y )
1614, 15syl6 34 . . . . 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 220 . . . 4  |-  ( P  =/=  Q  ->  (
( ( x  e. LinesEE  /\  y  e. LinesEE )  /\  ( ( P  e.  x  /\  Q  e.  x )  /\  ( P  e.  y  /\  Q  e.  y )
) )  ->  x  =  y ) )
1817expd 437 . . 3  |-  ( P  =/=  Q  ->  (
( x  e. LinesEE  /\  y  e. LinesEE )  ->  ( (
( P  e.  x  /\  Q  e.  x
)  /\  ( P  e.  y  /\  Q  e.  y ) )  ->  x  =  y )
) )
1918ralrimivv 2842 . 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 715 . . 3  |-  ( x  =  y  ->  (
( P  e.  x  /\  Q  e.  x
)  <->  ( P  e.  y  /\  Q  e.  y ) ) )
2322rmo4 3263 . 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 215 1  |-  ( P  =/=  Q  ->  E* x  e. LinesEE  ( P  e.  x  /\  Q  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E*wrmo 2774  (class class class)co 6305  Linecline2 30906  LinesEEclines2 30908
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-inf2 8155  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  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  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-se 4813  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-isom 5610  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-ec 7376  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-sum 13752  df-ee 24919  df-btwn 24920  df-cgr 24921  df-ofs 30755  df-colinear 30811  df-ifs 30812  df-cgr3 30813  df-fs 30814  df-line2 30909  df-lines2 30911
This theorem is referenced by:  linethrueu  30928
  Copyright terms: Public domain W3C validator