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Theorem usgn0fidegnn0 25750
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 3740 . . . 4  |-  ( V  =/=  (/)  <->  E. k  k  e.  V )
2 usgfidegfi 25631 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  A. u  e.  V  ( ( V VDeg  E ) `  u
)  e.  NN0 )
3 fveq2 5863 . . . . . . . . . . . 12  |-  ( u  =  k  ->  (
( V VDeg  E ) `  u )  =  ( ( V VDeg  E ) `
 k ) )
43eleq1d 2512 . . . . . . . . . . 11  |-  ( u  =  k  ->  (
( ( V VDeg  E
) `  u )  e.  NN0  <->  ( ( V VDeg 
E ) `  k
)  e.  NN0 )
)
54rspccv 3146 . . . . . . . . . 10  |-  ( A. u  e.  V  (
( V VDeg  E ) `  u )  e.  NN0  ->  ( k  e.  V  ->  ( ( V VDeg  E
) `  k )  e.  NN0 ) )
62, 5syl 17 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  (
k  e.  V  -> 
( ( V VDeg  E
) `  k )  e.  NN0 ) )
763impia 1204 . . . . . . . 8  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  k  e.  V )  ->  (
( V VDeg  E ) `  k )  e.  NN0 )
8 risset 2914 . . . . . . . . 9  |-  ( ( ( V VDeg  E ) `
 k )  e. 
NN0 
<->  E. n  e.  NN0  n  =  ( ( V VDeg  E ) `  k
) )
9 fveq2 5863 . . . . . . . . . . . . . . . 16  |-  ( v  =  k  ->  (
( V VDeg  E ) `  v )  =  ( ( V VDeg  E ) `
 k ) )
109eqeq1d 2452 . . . . . . . . . . . . . . 15  |-  ( v  =  k  ->  (
( ( V VDeg  E
) `  v )  =  n  <->  ( ( V VDeg 
E ) `  k
)  =  n ) )
11 eqcom 2457 . . . . . . . . . . . . . . 15  |-  ( ( ( V VDeg  E ) `
 k )  =  n  <->  n  =  (
( V VDeg  E ) `  k ) )
1210, 11syl6bb 265 . . . . . . . . . . . . . 14  |-  ( v  =  k  ->  (
( ( V VDeg  E
) `  v )  =  n  <->  n  =  (
( V VDeg  E ) `  k ) ) )
1312rexbidv 2900 . . . . . . . . . . . . 13  |-  ( v  =  k  ->  ( E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n  <->  E. n  e.  NN0  n  =  ( ( V VDeg  E ) `  k
) ) )
1413rspcev 3149 . . . . . . . . . . . 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 436 . . . . . . . . . . 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 1030 . . . . . . . . . 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 32 . . . . . . . . 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 199 . . . . . . . 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 37 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  k  e.  V )  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n )
20193exp 1206 . . . . . 6  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( k  e.  V  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n ) ) )
2120com13 83 . . . . 5  |-  ( k  e.  V  ->  ( V  e.  Fin  ->  ( V USGrph  E  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg 
E ) `  v
)  =  n ) ) )
2221exlimiv 1775 . . . 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 199 . . 3  |-  ( V  =/=  (/)  ->  ( V  e.  Fin  ->  ( V USGrph  E  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n ) ) )
2423com13 83 . 2  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  E. v  e.  V  E. n  e.  NN0  ( ( V VDeg  E
) `  v )  =  n ) ) )
25243imp 1201 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 371    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886    =/= wne 2621   A.wral 2736   E.wrex 2737   (/)c0 3730   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   Fincfn 7566   NN0cn0 10866   USGrph cusg 25050   VDeg cvdg 25614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-xadd 11407  df-fz 11782  df-hash 12513  df-usgra 25053  df-vdgr 25615
This theorem is referenced by:  friendshipgt3  25842
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