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Theorem vdgfrgragt2 28132
Description: Any vertex in a friendship graph (with more than one vertex - then, actually, the graph must have at least three vertices, because otherwise, it would not be a friendship graph) has at least degree 2, see 3. remark after Proposition 1 of [MertziosUnger] p. 153 : "It follows that deg(v) >= 2 for every node v of a friendship graph". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
Assertion
Ref Expression
vdgfrgragt2  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  (
1  <  ( # `  V
)  ->  2  <_  ( ( V VDeg  E ) `
 N ) ) )

Proof of Theorem vdgfrgragt2
StepHypRef Expression
1 vdgn0frgrav2 28129 . . . 4  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  (
1  <  ( # `  V
)  ->  ( ( V VDeg  E ) `  N
)  =/=  0 ) )
21imp 419 . . 3  |-  ( ( ( V FriendGrph  E  /\  N  e.  V )  /\  1  <  ( # `  V ) )  -> 
( ( V VDeg  E
) `  N )  =/=  0 )
3 vdgn1frgrav2 28131 . . . 4  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  (
1  <  ( # `  V
)  ->  ( ( V VDeg  E ) `  N
)  =/=  1 ) )
43imp 419 . . 3  |-  ( ( ( V FriendGrph  E  /\  N  e.  V )  /\  1  <  ( # `  V ) )  -> 
( ( V VDeg  E
) `  N )  =/=  1 )
5 frisusgra 28096 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
6 usgrav 21324 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
76simpld 446 . . . . . . . . . 10  |-  ( V USGrph  E  ->  V  e.  _V )
85, 7syl 16 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  V  e.  _V )
9 usgrafun 21331 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  Fun  E )
105, 9syl 16 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  Fun  E )
11 funfn 5441 . . . . . . . . . 10  |-  ( Fun 
E  <->  E  Fn  dom  E )
1210, 11sylib 189 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  E  Fn  dom  E )
136simprd 450 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  E  e.  _V )
145, 13syl 16 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  E  e.  _V )
15 dmexg 5089 . . . . . . . . . 10  |-  ( E  e.  _V  ->  dom  E  e.  _V )
1614, 15syl 16 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  dom  E  e.  _V )
178, 12, 163jca 1134 . . . . . . . 8  |-  ( V FriendGrph  E  ->  ( V  e. 
_V  /\  E  Fn  dom  E  /\  dom  E  e.  _V ) )
1817adantr 452 . . . . . . 7  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  ( V  e.  _V  /\  E  Fn  dom  E  /\  dom  E  e.  _V ) )
19 vdgrf 21622 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  Fn  dom  E  /\  dom  E  e.  _V )  ->  ( V VDeg  E ) : V --> ( NN0 
u.  {  +oo } ) )
2018, 19syl 16 . . . . . 6  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  ( V VDeg  E ) : V --> ( NN0  u.  {  +oo } ) )
21 simpr 448 . . . . . 6  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  N  e.  V )
2220, 21ffvelrnd 5830 . . . . 5  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  (
( V VDeg  E ) `  N )  e.  ( NN0  u.  {  +oo } ) )
23 elun 3448 . . . . . 6  |-  ( ( ( V VDeg  E ) `
 N )  e.  ( NN0  u.  {  +oo } )  <->  ( (
( V VDeg  E ) `  N )  e.  NN0  \/  ( ( V VDeg  E
) `  N )  e.  {  +oo } ) )
24 nn0n0n1ge2 10236 . . . . . . . 8  |-  ( ( ( ( V VDeg  E
) `  N )  e.  NN0  /\  ( ( V VDeg  E ) `  N )  =/=  0  /\  ( ( V VDeg  E
) `  N )  =/=  1 )  ->  2  <_  ( ( V VDeg  E
) `  N )
)
25243exp 1152 . . . . . . 7  |-  ( ( ( V VDeg  E ) `
 N )  e. 
NN0  ->  ( ( ( V VDeg  E ) `  N )  =/=  0  ->  ( ( ( V VDeg 
E ) `  N
)  =/=  1  -> 
2  <_  ( ( V VDeg  E ) `  N
) ) ) )
26 elsni 3798 . . . . . . . 8  |-  ( ( ( V VDeg  E ) `
 N )  e. 
{  +oo }  ->  (
( V VDeg  E ) `  N )  =  +oo )
27 2re 10025 . . . . . . . . . . . . 13  |-  2  e.  RR
2827rexri 9093 . . . . . . . . . . . 12  |-  2  e.  RR*
29 pnfge 10683 . . . . . . . . . . . 12  |-  ( 2  e.  RR*  ->  2  <_  +oo )
3028, 29ax-mp 8 . . . . . . . . . . 11  |-  2  <_  +oo
31 breq2 4176 . . . . . . . . . . 11  |-  ( ( ( V VDeg  E ) `
 N )  = 
+oo  ->  ( 2  <_ 
( ( V VDeg  E
) `  N )  <->  2  <_  +oo ) )
3230, 31mpbiri 225 . . . . . . . . . 10  |-  ( ( ( V VDeg  E ) `
 N )  = 
+oo  ->  2  <_  (
( V VDeg  E ) `  N ) )
3332a1d 23 . . . . . . . . 9  |-  ( ( ( V VDeg  E ) `
 N )  = 
+oo  ->  ( ( ( V VDeg  E ) `  N )  =/=  1  ->  2  <_  ( ( V VDeg  E ) `  N
) ) )
3433a1d 23 . . . . . . . 8  |-  ( ( ( V VDeg  E ) `
 N )  = 
+oo  ->  ( ( ( V VDeg  E ) `  N )  =/=  0  ->  ( ( ( V VDeg 
E ) `  N
)  =/=  1  -> 
2  <_  ( ( V VDeg  E ) `  N
) ) ) )
3526, 34syl 16 . . . . . . 7  |-  ( ( ( V VDeg  E ) `
 N )  e. 
{  +oo }  ->  (
( ( V VDeg  E
) `  N )  =/=  0  ->  ( ( ( V VDeg  E ) `
 N )  =/=  1  ->  2  <_  ( ( V VDeg  E ) `
 N ) ) ) )
3625, 35jaoi 369 . . . . . 6  |-  ( ( ( ( V VDeg  E
) `  N )  e.  NN0  \/  ( ( V VDeg  E ) `  N )  e.  {  +oo } )  ->  (
( ( V VDeg  E
) `  N )  =/=  0  ->  ( ( ( V VDeg  E ) `
 N )  =/=  1  ->  2  <_  ( ( V VDeg  E ) `
 N ) ) ) )
3723, 36sylbi 188 . . . . 5  |-  ( ( ( V VDeg  E ) `
 N )  e.  ( NN0  u.  {  +oo } )  ->  (
( ( V VDeg  E
) `  N )  =/=  0  ->  ( ( ( V VDeg  E ) `
 N )  =/=  1  ->  2  <_  ( ( V VDeg  E ) `
 N ) ) ) )
3822, 37syl 16 . . . 4  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  (
( ( V VDeg  E
) `  N )  =/=  0  ->  ( ( ( V VDeg  E ) `
 N )  =/=  1  ->  2  <_  ( ( V VDeg  E ) `
 N ) ) ) )
3938adantr 452 . . 3  |-  ( ( ( V FriendGrph  E  /\  N  e.  V )  /\  1  <  ( # `  V ) )  -> 
( ( ( V VDeg 
E ) `  N
)  =/=  0  -> 
( ( ( V VDeg 
E ) `  N
)  =/=  1  -> 
2  <_  ( ( V VDeg  E ) `  N
) ) ) )
402, 4, 39mp2d 43 . 2  |-  ( ( ( V FriendGrph  E  /\  N  e.  V )  /\  1  <  ( # `  V ) )  -> 
2  <_  ( ( V VDeg  E ) `  N
) )
4140ex 424 1  |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  (
1  <  ( # `  V
)  ->  2  <_  ( ( V VDeg  E ) `
 N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    u. cun 3278   {csn 3774   class class class wbr 4172   dom cdm 4837   Fun wfun 5407    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077   2c2 10005   NN0cn0 10177   #chash 11573   USGrph cusg 21318   VDeg cvdg 21617   FriendGrph cfrgra 28092
This theorem is referenced by:  frgrawopreglem2  28148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-xadd 10667  df-fz 11000  df-hash 11574  df-usgra 21320  df-vdgr 21618  df-frgra 28093
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