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Theorem segconeu 29238
Description: Existential uniqueness version of segconeq 29237. (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 457 . . 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 1003 . . 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 1002 . . 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 23906 . . 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 1228 . 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 1076 . . . . . . 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 755 . . . . . . 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 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 , 
s >.  /\  <. D , 
s >.Cgr <. A ,  B >. ) )
96, 7, 83jca 1176 . . . . . 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 434 . . . . 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 996 . . . . . 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 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 ) ) )  ->  D  e.  ( EE `  N ) )
13 simp21l 1113 . . . . . 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 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 ) ) )  ->  B  e.  ( EE `  N ) )
15 simp22l 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 ) ) )  ->  C  e.  ( EE `  N ) )
16 simp3l 1024 . . . . . 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 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 ) ) )  ->  s  e.  ( EE `  N ) )
18 segconeq 29237 . . . . . 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 1251 . . . . 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 44 . . . 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 1196 . . 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 2885 . 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 4214 . . . . 5  |-  ( r  =  s  ->  <. C , 
r >.  =  <. C , 
s >. )
2423breq2d 4459 . . . 4  |-  ( r  =  s  ->  ( D  Btwn  <. C ,  r
>. 
<->  D  Btwn  <. C , 
s >. ) )
25 opeq2 4214 . . . . 5  |-  ( r  =  s  ->  <. D , 
r >.  =  <. D , 
s >. )
2625breq1d 4457 . . . 4  |-  ( r  =  s  ->  ( <. D ,  r >.Cgr <. A ,  B >.  <->  <. D ,  s >.Cgr <. A ,  B >. ) )
2724, 26anbi12d 710 . . 3  |-  ( r  =  s  ->  (
( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  s >.  /\ 
<. D ,  s >.Cgr <. A ,  B >. ) ) )
2827reu4 3297 . 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 664 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 369    /\ w3a 973    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   <.cop 4033   class class class wbr 4447   ` cfv 5586   NNcn 10532   EEcee 23867    Btwn cbtwn 23868  Cgrccgr 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-ee 23870  df-btwn 23871  df-cgr 23872  df-ofs 29210
This theorem is referenced by:  transportcl  29260  transportprops  29261
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