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Theorem segconeu 24634
Description: Existential uniqueness version of segconeq 24633. (Contributed by Scott Fenton, 19-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
segconeu  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  ->  E! r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )
Distinct variable groups:    N, r    A, r    B, r    C, r    D, r

Proof of Theorem segconeu
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  ->  N  e.  NN )
2 simpr2 962 . . 3  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  -> 
( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )
3 simpr1 961 . . 3  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  -> 
( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )
4 axsegcon 24555 . . 3  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  E. r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )
51, 2, 3, 4syl3anc 1182 . 2  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  ->  E. r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )
6 simpl23 1035 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  /\  ( ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) ) )  ->  C  =/=  D )
7 simprl 732 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  /\  ( ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) ) )  -> 
( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )
8 simprr 733 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  /\  ( ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) ) )  -> 
( D  Btwn  <. C , 
s >.  /\  <. D , 
s >.Cgr <. A ,  B >. ) )
96, 7, 83jca 1132 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  /\  ( ( D 
Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) ) )  -> 
( C  =/=  D  /\  ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) ) )
109ex 423 . . . . 5  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  ( ( ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) )  ->  ( C  =/=  D  /\  ( D  Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) ) ) )
11 simp1 955 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  N  e.  NN )
12 simp22r 1075 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
13 simp21l 1072 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
14 simp21r 1073 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N ) )
15 simp22l 1074 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
16 simp3l 983 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  r  e.  ( EE `  N ) )
17 simp3r 984 . . . . . 6  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  s  e.  ( EE `  N ) )
18 segconeq 24633 . . . . . 6  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  r  e.  ( EE `  N
)  /\  s  e.  ( EE `  N ) ) )  ->  (
( C  =/=  D  /\  ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) )  ->  r  =  s ) )
1911, 12, 13, 14, 15, 16, 17, 18syl133anc 1205 . . . . 5  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  ( ( C  =/=  D  /\  ( D  Btwn  <. C ,  r
>.  /\  <. D ,  r
>.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) )  ->  r  =  s ) )
2010, 19syld 40 . . . 4  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D )  /\  (
r  e.  ( EE
`  N )  /\  s  e.  ( EE `  N ) ) )  ->  ( ( ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) )  ->  r  =  s ) )
21203expa 1151 . . 3  |-  ( ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  /\  ( r  e.  ( EE `  N )  /\  s  e.  ( EE `  N ) ) )  ->  (
( ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. )  /\  ( D  Btwn  <. C ,  s >.  /\ 
<. D ,  s >.Cgr <. A ,  B >. ) )  ->  r  =  s ) )
2221ralrimivva 2635 . 2  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  ->  A. r  e.  ( EE `  N ) A. s  e.  ( EE `  N ) ( ( ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) )  ->  r  =  s ) )
23 opeq2 3797 . . . . 5  |-  ( r  =  s  ->  <. C , 
r >.  =  <. C , 
s >. )
2423breq2d 4035 . . . 4  |-  ( r  =  s  ->  ( D  Btwn  <. C ,  r
>. 
<->  D  Btwn  <. C , 
s >. ) )
25 opeq2 3797 . . . . 5  |-  ( r  =  s  ->  <. D , 
r >.  =  <. D , 
s >. )
2625breq1d 4033 . . . 4  |-  ( r  =  s  ->  ( <. D ,  r >.Cgr <. A ,  B >.  <->  <. D ,  s >.Cgr <. A ,  B >. ) )
2724, 26anbi12d 691 . . 3  |-  ( r  =  s  ->  (
( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  s >.  /\ 
<. D ,  s >.Cgr <. A ,  B >. ) ) )
2827reu4 2959 . 2  |-  ( E! r  e.  ( EE
`  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  <->  ( E. r  e.  ( EE `  N
) ( D  Btwn  <. C ,  r >.  /\ 
<. D ,  r >.Cgr <. A ,  B >. )  /\  A. r  e.  ( EE `  N
) A. s  e.  ( EE `  N
) ( ( ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  /\  ( D 
Btwn  <. C ,  s
>.  /\  <. D ,  s
>.Cgr <. A ,  B >. ) )  ->  r  =  s ) ) )
295, 22, 28sylanbrc 645 1  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  C  =/=  D ) )  ->  E! r  e.  ( EE `  N ) ( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516    Btwn cbtwn 24517  Cgrccgr 24518
This theorem is referenced by:  transportcl  24656  transportprops  24657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ee 24519  df-btwn 24520  df-cgr 24521  df-ofs 24606
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