MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgn0fidegnn0 Structured version   Unicode version

Theorem usgn0fidegnn0 25156
Description: In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
Assertion
Ref Expression
usgn0fidegnn0  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n )
Distinct variable groups:    n, E, v    n, V, v

Proof of Theorem usgn0fidegnn0
Dummy variables  k  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3803 . . . 4  |-  ( V  =/=  (/)  <->  E. k  k  e.  V )
2 usgfidegfi 25037 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. u  e.  V  ( ( V VDeg  E ) `  u
)  e.  NN0 )
3 fveq2 5872 . . . . . . . . . . . 12  |-  ( u  =  k  ->  (
( V VDeg  E ) `  u )  =  ( ( V VDeg  E ) `
 k ) )
43eleq1d 2526 . . . . . . . . . . 11  |-  ( u  =  k  ->  (
( ( V VDeg  E
) `  u )  e.  NN0  <->  ( ( V VDeg 
E ) `  k
)  e.  NN0 )
)
54rspccv 3207 . . . . . . . . . 10  |-  ( A. u  e.  V  (
( V VDeg  E ) `  u )  e.  NN0  ->  ( k  e.  V  ->  ( ( V VDeg  E
) `  k )  e.  NN0 ) )
62, 5syl 16 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  (
k  e.  V  -> 
( ( V VDeg  E
) `  k )  e.  NN0 ) )
763impia 1193 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  k  e.  V )  ->  (
( V VDeg  E ) `  k )  e.  NN0 )
8 risset 2982 . . . . . . . . 9  |-  ( ( ( V VDeg  E ) `
 k )  e. 
NN0 
<->  E. n  e.  NN0  n  =  ( ( V VDeg  E ) `  k
) )
9 fveq2 5872 . . . . . . . . . . . . . . . 16  |-  ( v  =  k  ->  (
( V VDeg  E ) `  v )  =  ( ( V VDeg  E ) `
 k ) )
109eqeq1d 2459 . . . . . . . . . . . . . . 15  |-  ( v  =  k  ->  (
( ( V VDeg  E
) `  v )  =  n  <->  ( ( V VDeg 
E ) `  k
)  =  n ) )
11 eqcom 2466 . . . . . . . . . . . . . . 15  |-  ( ( ( V VDeg  E ) `
 k )  =  n  <->  n  =  (
( V VDeg  E ) `  k ) )
1210, 11syl6bb 261 . . . . . . . . . . . . . 14  |-  ( v  =  k  ->  (
( ( V VDeg  E
) `  v )  =  n  <->  n  =  (
( V VDeg  E ) `  k ) ) )
1312rexbidv 2968 . . . . . . . . . . . . 13  |-  ( v  =  k  ->  ( E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n  <->  E. n  e.  NN0  n  =  ( ( V VDeg  E ) `  k
) ) )
1413rspcev 3210 . . . . . . . . . . . 12  |-  ( ( k  e.  V  /\  E. n  e.  NN0  n  =  ( ( V VDeg 
E ) `  k
) )  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n )
1514ex 434 . . . . . . . . . . 11  |-  ( k  e.  V  ->  ( E. n  e.  NN0  n  =  ( ( V VDeg  E ) `  k
)  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n ) )
16153ad2ant3 1019 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  k  e.  V )  ->  ( E. n  e.  NN0  n  =  ( ( V VDeg  E ) `  k
)  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n ) )
1716com12 31 . . . . . . . . 9  |-  ( E. n  e.  NN0  n  =  ( ( V VDeg 
E ) `  k
)  ->  ( ( V USGrph  E  /\  V  e. 
Fin  /\  k  e.  V )  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n ) )
188, 17sylbi 195 . . . . . . . 8  |-  ( ( ( V VDeg  E ) `
 k )  e. 
NN0  ->  ( ( V USGrph  E  /\  V  e.  Fin  /\  k  e.  V )  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n ) )
197, 18mpcom 36 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  k  e.  V )  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n )
20193exp 1195 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( k  e.  V  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n ) ) )
2120com13 80 . . . . 5  |-  ( k  e.  V  ->  ( V  e.  Fin  ->  ( V USGrph  E  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n ) ) )
2221exlimiv 1723 . . . 4  |-  ( E. k  k  e.  V  ->  ( V  e.  Fin  ->  ( V USGrph  E  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n ) ) )
231, 22sylbi 195 . . 3  |-  ( V  =/=  (/)  ->  ( V  e.  Fin  ->  ( V USGrph  E  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n ) ) )
2423com13 80 . 2  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n ) ) )
25243imp 1190 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   (/)c0 3793   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Fincfn 7535   NN0cn0 10816   USGrph cusg 24457   VDeg cvdg 25020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-xadd 11344  df-fz 11698  df-hash 12409  df-usgra 24460  df-vdgr 25021
This theorem is referenced by:  friendshipgt3  25248
  Copyright terms: Public domain W3C validator