MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgra2edg Structured version   Unicode version

Theorem usgra2edg 24074
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 24041 . . . . . 6  |-  ( V USGrph  E  ->  Fun  E )
2 funfn 5616 . . . . . 6  |-  ( Fun 
E  <->  E  Fn  dom  E )
31, 2sylib 196 . . . . 5  |-  ( V USGrph  E  ->  E  Fn  dom  E )
433ad2ant1 1017 . . . 4  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  E  Fn  dom  E )
5 fvelrnb 5914 . . . . 5  |-  ( E  Fn  dom  E  -> 
( { N , 
b }  e.  ran  E  <->  E. x  e.  dom  E ( E `  x
)  =  { N ,  b } ) )
6 fvelrnb 5914 . . . . 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 3029 . . . 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 5865 . . . . . . . . . . . . . . 15  |-  ( x  =  y  ->  ( E `  x )  =  ( E `  y ) )
1110eqeq1d 2469 . . . . . . . . . . . . . 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 2494 . . . . . . . . . . . . . 14  |-  ( ( ( E `  y
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  ->  { N ,  b }  =  { c ,  N } )
14 prcom 4105 . . . . . . . . . . . . . . . 16  |-  { c ,  N }  =  { N ,  c }
1514eqeq2i 2485 . . . . . . . . . . . . . . 15  |-  ( { N ,  b }  =  { c ,  N }  <->  { N ,  b }  =  { N ,  c } )
16 vex 3116 . . . . . . . . . . . . . . . . 17  |-  b  e. 
_V
17 vex 3116 . . . . . . . . . . . . . . . . 17  |-  c  e. 
_V
1816, 17preqr2 4201 . . . . . . . . . . . . . . . 16  |-  ( { N ,  b }  =  { N , 
c }  ->  b  =  c )
19 df-ne 2664 . . . . . . . . . . . . . . . . 17  |-  ( b  =/=  c  <->  -.  b  =  c )
20 pm2.24 109 . . . . . . . . . . . . . . . . 17  |-  ( b  =  c  ->  ( -.  b  =  c  ->  x  =/=  y ) )
2119, 20syl5bi 217 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
b  =/=  c  ->  x  =/=  y ) )
2218, 21syl 16 . . . . . . . . . . . . . . 15  |-  ( { N ,  b }  =  { N , 
c }  ->  (
b  =/=  c  ->  x  =/=  y ) )
2315, 22sylbi 195 . . . . . . . . . . . . . 14  |-  ( { N ,  b }  =  { c ,  N }  ->  (
b  =/=  c  ->  x  =/=  y ) )
2413, 23syl 16 . . . . . . . . . . . . 13  |-  ( ( ( E `  y
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( b  =/=  c  ->  x  =/=  y ) )
2512, 24syl6bi 228 . . . . . . . . . . . 12  |-  ( x  =  y  ->  (
( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } )  ->  ( b  =/=  c  ->  x  =/=  y ) ) )
2625com23 78 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
b  =/=  c  -> 
( ( ( E `
 x )  =  { N ,  b }  /\  ( E `
 y )  =  { c ,  N } )  ->  x  =/=  y ) ) )
2726impd 431 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y ) )
28 ax-1 6 . . . . . . . . . 10  |-  ( x  =/=  y  ->  (
( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y ) )
2927, 28pm2.61ine 2780 . . . . . . . . 9  |-  ( ( b  =/=  c  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y )
30293ad2antl3 1160 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  x  =/=  y )
31 prid1g 4133 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  N  e.  { N ,  b } )
32313ad2ant2 1018 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  { N ,  b } )
33 eleq2 2540 . . . . . . . . . . 11  |-  ( ( E `  x )  =  { N , 
b }  ->  ( N  e.  ( E `  x )  <->  N  e.  { N ,  b } ) )
3432, 33syl5ibr 221 . . . . . . . . . 10  |-  ( ( E `  x )  =  { N , 
b }  ->  (
( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  ( E `  x ) ) )
3534adantr 465 . . . . . . . . 9  |-  ( ( ( E `  x
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
)  ->  N  e.  ( E `  x ) ) )
3635impcom 430 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  N  e.  ( E `  x ) )
37 prid2g 4134 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  N  e.  { c ,  N } )
38373ad2ant2 1018 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  { c ,  N } )
39 eleq2 2540 . . . . . . . . . . 11  |-  ( ( E `  y )  =  { c ,  N }  ->  ( N  e.  ( E `  y )  <->  N  e.  { c ,  N }
) )
4038, 39syl5ibr 221 . . . . . . . . . 10  |-  ( ( E `  y )  =  { c ,  N }  ->  (
( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  ->  N  e.  ( E `  y ) ) )
4140adantl 466 . . . . . . . . 9  |-  ( ( ( E `  x
)  =  { N ,  b }  /\  ( E `  y )  =  { c ,  N } )  -> 
( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c
)  ->  N  e.  ( E `  y ) ) )
4241impcom 430 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  N  e.  V  /\  b  =/=  c )  /\  ( ( E `  x )  =  { N ,  b }  /\  ( E `  y
)  =  { c ,  N } ) )  ->  N  e.  ( E `  y ) )
4330, 36, 423jca 1176 . . . . . . 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 ) ) )
4443ex 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 ) ) ) )
4544reximdv 2937 . . . . 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 )
) ) )
4645reximdv 2937 . . . 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 )
) ) )
479, 46syl5bir 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 ) ) ) )
488, 47sylbid 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 ) ) ) )
4948imp 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:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {cpr 4029   class class class wbr 4447   dom cdm 4999   ran crn 5000   Fun wfun 5581    Fn wfn 5582   ` cfv 5587   USGrph cusg 24022
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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-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-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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-hash 12373  df-usgra 24025
This theorem is referenced by:  usgra2edg1  24075
  Copyright terms: Public domain W3C validator