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Theorem segconeu 25937
Description: Existential uniqueness version of segconeq 25936. (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 444 . . 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 964 . . 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 963 . . 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 25858 . . 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 1184 . 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 1037 . . . . . . 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 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 , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )
8 simprr 734 . . . . . . 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 1134 . . . . . 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 424 . . . . 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 957 . . . . . 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 1077 . . . . . 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 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 ) ) )  ->  A  e.  ( EE `  N ) )
14 simp21r 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 ) ) )  ->  B  e.  ( EE `  N ) )
15 simp22l 1076 . . . . . 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 985 . . . . . 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 986 . . . . . 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 25936 . . . . . 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 1207 . . . . 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 42 . . . 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 1153 . . 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 2790 . 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 3977 . . . . 5  |-  ( r  =  s  ->  <. C , 
r >.  =  <. C , 
s >. )
2423breq2d 4216 . . . 4  |-  ( r  =  s  ->  ( D  Btwn  <. C ,  r
>. 
<->  D  Btwn  <. C , 
s >. ) )
25 opeq2 3977 . . . . 5  |-  ( r  =  s  ->  <. D , 
r >.  =  <. D , 
s >. )
2625breq1d 4214 . . . 4  |-  ( r  =  s  ->  ( <. D ,  r >.Cgr <. A ,  B >.  <->  <. D ,  s >.Cgr <. A ,  B >. ) )
2724, 26anbi12d 692 . . 3  |-  ( r  =  s  ->  (
( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  s >.  /\ 
<. D ,  s >.Cgr <. A ,  B >. ) ) )
2827reu4 3120 . 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 646 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 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   E!wreu 2699   <.cop 3809   class class class wbr 4204   ` cfv 5446   NNcn 9992   EEcee 25819    Btwn cbtwn 25820  Cgrccgr 25821
This theorem is referenced by:  transportcl  25959  transportprops  25960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-ee 25822  df-btwn 25823  df-cgr 25824  df-ofs 25909
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