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Theorem usgra2edg 24360
Description: If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
Assertion
Ref Expression
usgra2edg  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( { N ,  b }  e.  ran  E  /\  { c ,  N }  e.  ran  E ) )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) )
Distinct variable groups:    E, b,
c, x, y    N, b, c, x, y    x, V, y
Allowed substitution hints:    V( b, c)

Proof of Theorem usgra2edg
StepHypRef Expression
1 usgrafun 24327 . . . . . 6  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5607 . . . . . 6  |-  ( Fun 
E  <->  E  Fn  dom  E )
31, 2sylib 196 . . . . 5  |-  ( V USGrph  E  ->  E  Fn  dom  E )
433ad2ant1 1018 . . . 4  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  E  Fn  dom  E )
5 fvelrnb 5905 . . . . 5  |-  ( E  Fn  dom  E  -> 
( { N , 
b }  e.  ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  { N ,  b } ) )
6 fvelrnb 5905 . . . . 5  |-  ( E  Fn  dom  E  -> 
( { c ,  N }  e.  ran  E  <->  E. y  e.  dom  E ( E `  y
)  =  { c ,  N } ) )
75, 6anbi12d 710 . . . 4  |-  ( E  Fn  dom  E  -> 
( ( { N ,  b }  e.  ran  E  /\  { c ,  N }  e.  ran  E )  <->  ( E. x  e.  dom  E ( E `  x )  =  { N , 
b }  /\  E. y  e.  dom  E ( E `  y )  =  { c ,  N } ) ) )
84, 7syl 16 . . 3  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( { N , 
b }  e.  ran  E  /\  { c ,  N }  e.  ran  E )  <->  ( E. x  e.  dom  E ( E `
 x )  =  { N ,  b }  /\  E. y  e.  dom  E ( E `
 y )  =  { c ,  N } ) ) )
9 reeanv 3011 . . . 4  |-  ( E. x  e.  dom  E E. y  e.  dom  E ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  <-> 
( E. x  e. 
dom  E ( E `
 x )  =  { N ,  b }  /\  E. y  e.  dom  E ( E `
 y )  =  { c ,  N } ) )
10 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( E `  x )  =  ( E `  y ) )
1110eqeq1d 2445 . . . . . . . . . . . . . 14  |-  ( x  =  y  ->  (
( E `  x
)  =  { N ,  b }  <->  ( E `  y )  =  { N ,  b }
) )
1211anbi1d 704 . . . . . . . . . . . . 13  |-  ( x  =  y  ->  (
( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  <-> 
( ( E `  y )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) ) )
13 eqtr2 2470 . . . . . . . . . . . . . 14  |-  ( ( ( E `  y
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  ->  { N ,  b }  =  { c ,  N } )
14 prcom 4093 . . . . . . . . . . . . . . . 16  |-  { c ,  N }  =  { N ,  c }
1514eqeq2i 2461 . . . . . . . . . . . . . . 15  |-  ( { N ,  b }  =  { c ,  N }  <->  { N ,  b }  =  { N ,  c } )
16 vex 3098 . . . . . . . . . . . . . . . . 17  |-  b  e. 
_V
17 vex 3098 . . . . . . . . . . . . . . . . 17  |-  c  e. 
_V
1816, 17preqr2 4190 . . . . . . . . . . . . . . . 16  |-  ( { N ,  b }  =  { N , 
c }  ->  b  =  c )
19 eqneqall 2650 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
b  =/=  c  ->  x  =/=  y ) )
2018, 19syl 16 . . . . . . . . . . . . . . 15  |-  ( { N ,  b }  =  { N , 
c }  ->  (
b  =/=  c  ->  x  =/=  y ) )
2115, 20sylbi 195 . . . . . . . . . . . . . 14  |-  ( { N ,  b }  =  { c ,  N }  ->  (
b  =/=  c  ->  x  =/=  y ) )
2213, 21syl 16 . . . . . . . . . . . . 13  |-  ( ( ( E `  y
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( b  =/=  c  ->  x  =/=  y ) )
2312, 22syl6bi 228 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  ( b  =/=  c  ->  x  =/=  y ) ) )
2423com23 78 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
b  =/=  c  -> 
( ( ( E `
 x )  =  { N ,  b }  /\  ( E `
 y )  =  { c ,  N } )  ->  x  =/=  y ) ) )
2524impd 431 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y ) )
26 ax-1 6 . . . . . . . . . 10  |-  ( x  =/=  y  ->  (
( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y ) )
2725, 26pm2.61ine 2756 . . . . . . . . 9  |-  ( ( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y )
28273ad2antl3 1161 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y )
29 prid1g 4121 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  N  e.  { N ,  b } )
30293ad2ant2 1019 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  { N ,  b } )
31 eleq2 2516 . . . . . . . . . . 11  |-  ( ( E `  x )  =  { N , 
b }  ->  ( N  e.  ( E `  x )  <->  N  e.  { N ,  b } ) )
3230, 31syl5ibr 221 . . . . . . . . . 10  |-  ( ( E `  x )  =  { N , 
b }  ->  (
( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  ( E `  x ) ) )
3332adantr 465 . . . . . . . . 9  |-  ( ( ( E `  x
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
)  ->  N  e.  ( E `  x ) ) )
3433impcom 430 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  N  e.  ( E `  x ) )
35 prid2g 4122 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  N  e.  { c ,  N } )
36353ad2ant2 1019 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  { c ,  N } )
37 eleq2 2516 . . . . . . . . . . 11  |-  ( ( E `  y )  =  { c ,  N }  ->  ( N  e.  ( E `  y )  <->  N  e.  { c ,  N }
) )
3836, 37syl5ibr 221 . . . . . . . . . 10  |-  ( ( E `  y )  =  { c ,  N }  ->  (
( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  ( E `  y ) ) )
3938adantl 466 . . . . . . . . 9  |-  ( ( ( E `  x
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
)  ->  N  e.  ( E `  y ) ) )
4039impcom 430 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  N  e.  ( E `  y ) )
4128, 34, 403jca 1177 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) )
4241ex 434 . . . . . 6  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) ) )
4342reximdv 2917 . . . . 5  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  ( E. y  e.  dom  E ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x )  /\  N  e.  ( E `  y )
) ) )
4443reximdv 2917 . . . 4  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  ( E. x  e.  dom  E E. y  e.  dom  E ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x )  /\  N  e.  ( E `  y )
) ) )
459, 44syl5bir 218 . . 3  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( E. x  e. 
dom  E ( E `
 x )  =  { N ,  b }  /\  E. y  e.  dom  E ( E `
 y )  =  { c ,  N } )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) ) )
468, 45sylbid 215 . 2  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  (
( { N , 
b }  e.  ran  E  /\  { c ,  N }  e.  ran  E )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) ) )
4746imp 429 1  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( { N ,  b }  e.  ran  E  /\  { c ,  N }  e.  ran  E ) )  ->  E. x  e.  dom  E E. y  e.  dom  E ( x  =/=  y  /\  N  e.  ( E `  x
)  /\  N  e.  ( E `  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   {cpr 4016   class class class wbr 4437   dom cdm 4989   ran crn 4990   Fun wfun 5572    Fn wfn 5573   ` cfv 5578   USGrph cusg 24308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388  df-usgra 24311
This theorem is referenced by:  usgra2edg1  24361
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