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Theorem btwnswapid 25863
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 444 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
2 simpr2 964 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
3 simpr1 963 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
4 simpr3 965 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
5 axpasch 25792 . . 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 1202 . 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 731 . . . . . 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 448 . . . . . 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 999 . . . . . 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 25790 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  ->  (
x  Btwn  <. A ,  A >.  ->  x  =  A ) )
117, 8, 9, 10syl3anc 1184 . . . . 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 1000 . . . . . 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 25790 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  ->  (
x  Btwn  <. B ,  B >.  ->  x  =  B ) )
147, 8, 12, 13syl3anc 1184 . . . . 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 547 . . . 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 2430 . . . 4  |-  ( ( x  =  A  /\  x  =  B )  ->  A  =  B )
1715, 16syl6 31 . . 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 2798 . 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 42 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   <.cop 3785   class class class wbr 4180   ` cfv 5421   NNcn 9964   EEcee 25739    Btwn cbtwn 25740
This theorem is referenced by:  btwnswapid2  25864  segleantisym  25961  broutsideof2  25968
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-z 10247  df-uz 10453  df-icc 10887  df-fz 11008  df-ee 25742  df-btwn 25743
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