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Theorem segconeu 25950
Description: Existential uniqueness version of segconeq 25949. (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 445 . . 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 965 . . 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 964 . . 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 25871 . . 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 1185 . 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 1038 . . . . . . 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 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 , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. ) )
8 simprr 735 . . . . . . 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 1135 . . . . . 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 425 . . . . 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 958 . . . . . 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 1078 . . . . . 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 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 ) ) )  ->  A  e.  ( EE `  N ) )
14 simp21r 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 ) ) )  ->  B  e.  ( EE `  N ) )
15 simp22l 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 ) ) )  ->  C  e.  ( EE `  N ) )
16 simp3l 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 ) ) )  ->  r  e.  ( EE `  N ) )
17 simp3r 987 . . . . . 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 25949 . . . . . 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 1208 . . . . 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 43 . . . 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 1154 . . 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 2800 . 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 3987 . . . . 5  |-  ( r  =  s  ->  <. C , 
r >.  =  <. C , 
s >. )
2423breq2d 4227 . . . 4  |-  ( r  =  s  ->  ( D  Btwn  <. C ,  r
>. 
<->  D  Btwn  <. C , 
s >. ) )
25 opeq2 3987 . . . . 5  |-  ( r  =  s  ->  <. D , 
r >.  =  <. D , 
s >. )
2625breq1d 4225 . . . 4  |-  ( r  =  s  ->  ( <. D ,  r >.Cgr <. A ,  B >.  <->  <. D ,  s >.Cgr <. A ,  B >. ) )
2724, 26anbi12d 693 . . 3  |-  ( r  =  s  ->  (
( D  Btwn  <. C , 
r >.  /\  <. D , 
r >.Cgr <. A ,  B >. )  <->  ( D  Btwn  <. C ,  s >.  /\ 
<. D ,  s >.Cgr <. A ,  B >. ) ) )
2827reu4 3130 . 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 647 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 360    /\ w3a 937    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   E!wreu 2709   <.cop 3819   class class class wbr 4215   ` cfv 5457   NNcn 10005   EEcee 25832    Btwn cbtwn 25833  Cgrccgr 25834
This theorem is referenced by:  transportcl  25972  transportprops  25973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-ee 25835  df-btwn 25836  df-cgr 25837  df-ofs 25922
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