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Theorem btwnswapid 25952
Description: If you can swap the first two arguments of a betweenness statement, then those arguments are identical. Theorem 3.4 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
btwnswapid  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  /\  B  Btwn  <. A ,  C >. )  ->  A  =  B ) )

Proof of Theorem btwnswapid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 445 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
2 simpr2 965 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
3 simpr1 964 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
4 simpr3 966 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
5 axpasch 25881 . . 3  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  /\  B  Btwn  <. A ,  C >. )  ->  E. x  e.  ( EE `  N
) ( x  Btwn  <. A ,  A >.  /\  x  Btwn  <. B ,  B >. ) ) )
61, 2, 3, 4, 3, 2, 5syl132anc 1203 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  /\  B  Btwn  <. A ,  C >. )  ->  E. x  e.  ( EE `  N
) ( x  Btwn  <. A ,  A >.  /\  x  Btwn  <. B ,  B >. ) ) )
7 simpll 732 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  N  e.  NN )
8 simpr 449 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  x  e.  ( EE `  N
) )
9 simplr1 1000 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  A  e.  ( EE `  N
) )
10 axbtwnid 25879 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  ->  (
x  Btwn  <. A ,  A >.  ->  x  =  A ) )
117, 8, 9, 10syl3anc 1185 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  (
x  Btwn  <. A ,  A >.  ->  x  =  A ) )
12 simplr2 1001 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  B  e.  ( EE `  N
) )
13 axbtwnid 25879 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  (
x  Btwn  <. B ,  B >.  ->  x  =  B ) )
147, 8, 12, 13syl3anc 1185 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  (
x  Btwn  <. B ,  B >.  ->  x  =  B ) )
1511, 14anim12d 548 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  (
( x  Btwn  <. A ,  A >.  /\  x  Btwn  <. B ,  B >. )  ->  ( x  =  A  /\  x  =  B ) ) )
16 eqtr2 2455 . . . 4  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1715, 16syl6 32 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  x  e.  ( EE `  N
) )  ->  (
( x  Btwn  <. A ,  A >.  /\  x  Btwn  <. B ,  B >. )  ->  A  =  B ) )
1817rexlimdva 2831 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( E. x  e.  ( EE `  N
) ( x  Btwn  <. A ,  A >.  /\  x  Btwn  <. B ,  B >. )  ->  A  =  B ) )
196, 18syld 43 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  /\  B  Btwn  <. A ,  C >. )  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707   <.cop 3818   class class class wbr 4213   ` cfv 5455   NNcn 10001   EEcee 25828    Btwn cbtwn 25829
This theorem is referenced by:  btwnswapid2  25953  segleantisym  26050  broutsideof2  26057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-z 10284  df-uz 10490  df-icc 10924  df-fz 11045  df-ee 25831  df-btwn 25832
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