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Theorem segconeu 30336
Description: Existential uniqueness version of segconeq 30335. (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 455 . . 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 1004 . . 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 1003 . . 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 24634 . . 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 1230 . 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 1077 . . . . . . 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 756 . . . . . . 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 758 . . . . . . 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 1177 . . . . . 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 432 . . . . 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 997 . . . . . 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 1117 . . . . . 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 1114 . . . . . 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 1115 . . . . . 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 1116 . . . . . 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 1025 . . . . . 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 1026 . . . . . 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 30335 . . . . . 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 1253 . . . . 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 1197 . . 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 2824 . 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 4159 . . . . 5  |-  ( r  =  s  ->  <. C , 
r >.  =  <. C , 
s >. )
2423breq2d 4406 . . . 4  |-  ( r  =  s  ->  ( D  Btwn  <. C ,  r
>. 
<->  D  Btwn  <. C , 
s >. ) )
25 opeq2 4159 . . . . 5  |-  ( r  =  s  ->  <. D , 
r >.  =  <. D , 
s >. )
2625breq1d 4404 . . . 4  |-  ( r  =  s  ->  ( <. D ,  r >.Cgr <. A ,  B >.  <->  <. D ,  s >.Cgr <. A ,  B >. ) )
2724, 26anbi12d 709 . . 3  |-  ( r  =  s  ->  (
( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  s >.  /\ 
<. D ,  s >.Cgr <. A ,  B >. ) ) )
2827reu4 3242 . 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 662 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   E!wreu 2755   <.cop 3977   class class class wbr 4394   ` cfv 5568   NNcn 10575   EEcee 24595    Btwn cbtwn 24596  Cgrccgr 24597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-ee 24598  df-btwn 24599  df-cgr 24600  df-ofs 30308
This theorem is referenced by:  transportcl  30358  transportprops  30359
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