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Theorem nbhashuvtx1 25692
Description: If the number of the neighbors of a vertex in a finite graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
Assertion
Ref Expression
nbhashuvtx1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) )

Proof of Theorem nbhashuvtx1
StepHypRef Expression
1 ax-1 6 . . 3  |-  ( M  e.  ( <. V ,  E >. Neighbors  N )  ->  (
( M  e.  V  /\  M  =/=  N
)  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) )
212a1d 27 . 2  |-  ( M  e.  ( <. V ,  E >. Neighbors  N )  ->  (
( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) ) )
3 df-nel 2636 . . 3  |-  ( M  e/  ( <. V ,  E >. Neighbors  N )  <->  -.  M  e.  ( <. V ,  E >. Neighbors  N ) )
4 nbgrassvwo2 25215 . . . . . . 7  |-  ( ( V USGrph  E  /\  M  e/  ( <. V ,  E >. Neighbors  N ) )  -> 
( <. V ,  E >. Neighbors  N )  C_  ( V  \  { M ,  N } ) )
54ex 440 . . . . . 6  |-  ( V USGrph  E  ->  ( M  e/  ( <. V ,  E >. Neighbors  N )  ->  ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { M ,  N } ) ) )
653ad2ant1 1035 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( M  e/  ( <. V ,  E >. Neighbors  N )  ->  ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { M ,  N } ) ) )
7 difexg 4565 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  { M ,  N } )  e.  _V )
873ad2ant2 1036 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( V  \  { M ,  N } )  e.  _V )
9 hashss 12618 . . . . . . 7  |-  ( ( ( V  \  { M ,  N }
)  e.  _V  /\  ( <. V ,  E >. Neighbors  N )  C_  ( V  \  { M ,  N } ) )  -> 
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  ( # `
 ( V  \  { M ,  N }
) ) )
109ex 440 . . . . . 6  |-  ( ( V  \  { M ,  N } )  e. 
_V  ->  ( ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { M ,  N } )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  <_ 
( # `  ( V 
\  { M ,  N } ) ) ) )
118, 10syl 17 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( <. V ,  E >. Neighbors  N )  C_  ( V  \  { M ,  N } )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  <_ 
( # `  ( V 
\  { M ,  N } ) ) ) )
12 simpl2 1018 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  V  e.  Fin )
13 simpl 463 . . . . . . . . . . . . 13  |-  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  V )
14 simp3 1016 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  N  e.  V )
15 prssi 4141 . . . . . . . . . . . . 13  |-  ( ( M  e.  V  /\  N  e.  V )  ->  { M ,  N }  C_  V )
1613, 14, 15syl2anr 485 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  { M ,  N }  C_  V )
17 hashssdif 12621 . . . . . . . . . . . 12  |-  ( ( V  e.  Fin  /\  { M ,  N }  C_  V )  ->  ( # `
 ( V  \  { M ,  N }
) )  =  ( ( # `  V
)  -  ( # `  { M ,  N } ) ) )
1812, 16, 17syl2anc 671 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( V  \  { M ,  N } ) )  =  ( ( # `  V
)  -  ( # `  { M ,  N } ) ) )
19 simprr 771 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  M  =/=  N
)
20 hashprg 12604 . . . . . . . . . . . . . 14  |-  ( ( M  e.  V  /\  N  e.  V )  ->  ( M  =/=  N  <->  (
# `  { M ,  N } )  =  2 ) )
2113, 14, 20syl2anr 485 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( M  =/= 
N  <->  ( # `  { M ,  N }
)  =  2 ) )
2219, 21mpbid 215 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  { M ,  N }
)  =  2 )
2322oveq2d 6331 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  V )  -  ( # `
 { M ,  N } ) )  =  ( ( # `  V
)  -  2 ) )
2418, 23eqtrd 2496 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( V  \  { M ,  N } ) )  =  ( ( # `  V
)  -  2 ) )
2524breq2d 4428 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  <_ 
( # `  ( V 
\  { M ,  N } ) )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( # `  V )  -  2 ) ) )
26 nbhashnn0 25691 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e. 
NN0 )
2726nn0zd 11067 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e.  ZZ )
28 hashcl 12570 . . . . . . . . . . . . . . 15  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
29 nn0z 10989 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  e.  NN0  ->  ( # `  V
)  e.  ZZ )
30 peano2zm 11009 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  e.  ZZ  ->  ( ( # `
 V )  - 
1 )  e.  ZZ )
3128, 29, 303syl 18 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  -  1 )  e.  ZZ )
32313ad2ant2 1036 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  V )  -  1 )  e.  ZZ )
33 zltlem1 11018 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( <. V ,  E >. Neighbors  N
) )  e.  ZZ  /\  ( ( # `  V
)  -  1 )  e.  ZZ )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( ( # `  V
)  -  1 )  -  1 ) ) )
3427, 32, 33syl2anc 671 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( ( # `  V
)  -  1 )  -  1 ) ) )
3528nn0cnd 10956 . . . . . . . . . . . . . . . 16  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  CC )
36353ad2ant2 1036 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 V )  e.  CC )
37 1cnd 9685 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  1  e.  CC )
3836, 37, 37subsub4d 10043 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( ( # `  V
)  -  1 )  -  1 )  =  ( ( # `  V
)  -  ( 1  +  1 ) ) )
39 1p1e2 10751 . . . . . . . . . . . . . . 15  |-  ( 1  +  1 )  =  2
4039oveq2i 6326 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  -  ( 1  +  1 ) )  =  ( ( # `  V
)  -  2 )
4138, 40syl6eq 2512 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( ( # `  V
)  -  1 )  -  1 )  =  ( ( # `  V
)  -  2 ) )
4241breq2d 4428 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( ( # `  V
)  -  1 )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( # `  V )  -  2 ) ) )
4334, 42bitrd 261 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( # `  V )  -  2 ) ) )
4443adantr 471 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  < 
( ( # `  V
)  -  1 )  <-> 
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( # `  V )  -  2 ) ) )
4526nn0red 10955 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e.  RR )
4645adantr 471 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( <. V ,  E >. Neighbors  N
) )  e.  RR )
4746adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/= 
N ) )  /\  ( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 ) )  ->  ( # `  ( <. V ,  E >. Neighbors  N
) )  e.  RR )
48 simpr 467 . . . . . . . . . . . . 13  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/= 
N ) )  /\  ( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 ) )  ->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 ) )
4947, 48ltned 9797 . . . . . . . . . . . 12  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/= 
N ) )  /\  ( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 ) )  ->  ( # `  ( <. V ,  E >. Neighbors  N
) )  =/=  (
( # `  V )  -  1 ) )
5049ex 440 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  < 
( ( # `  V
)  -  1 )  ->  ( # `  ( <. V ,  E >. Neighbors  N
) )  =/=  (
( # `  V )  -  1 ) ) )
51 eqneqall 2646 . . . . . . . . . . . 12  |-  ( (
# `  ( <. V ,  E >. Neighbors  N ) )  =  ( (
# `  V )  -  1 )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  =/=  (
( # `  V )  -  1 )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) )
5251com12 32 . . . . . . . . . . 11  |-  ( (
# `  ( <. V ,  E >. Neighbors  N ) )  =/=  ( (
# `  V )  -  1 )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  M  e.  (
<. V ,  E >. Neighbors  N
) ) )
5350, 52syl6 34 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  < 
( ( # `  V
)  -  1 )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  =  ( ( # `  V
)  -  1 )  ->  M  e.  (
<. V ,  E >. Neighbors  N
) ) ) )
5444, 53sylbird 243 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  <_ 
( ( # `  V
)  -  2 )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  =  ( ( # `  V
)  -  1 )  ->  M  e.  (
<. V ,  E >. Neighbors  N
) ) ) )
5525, 54sylbid 223 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  ( <. V ,  E >. Neighbors  N ) )  <_ 
( # `  ( V 
\  { M ,  N } ) )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  M  e.  (
<. V ,  E >. Neighbors  N
) ) ) )
5655ex 440 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( M  e.  V  /\  M  =/=  N
)  ->  ( ( # `
 ( <. V ,  E >. Neighbors  N ) )  <_ 
( # `  ( V 
\  { M ,  N } ) )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  M  e.  (
<. V ,  E >. Neighbors  N
) ) ) ) )
5756com23 81 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  ( # `
 ( V  \  { M ,  N }
) )  ->  (
( M  e.  V  /\  M  =/=  N
)  ->  ( ( # `
 ( <. V ,  E >. Neighbors  N ) )  =  ( ( # `  V
)  -  1 )  ->  M  e.  (
<. V ,  E >. Neighbors  N
) ) ) ) )
5857com34 86 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  ( # `
 ( V  \  { M ,  N }
) )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) ) )
596, 11, 583syld 57 . . . 4  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( M  e/  ( <. V ,  E >. Neighbors  N )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) ) )
6059com12 32 . . 3  |-  ( M  e/  ( <. V ,  E >. Neighbors  N )  ->  (
( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) ) )
613, 60sylbir 218 . 2  |-  ( -.  M  e.  ( <. V ,  E >. Neighbors  N
)  ->  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) ) )
622, 61pm2.61i 169 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633    e/ wnel 2634   _Vcvv 3057    \ cdif 3413    C_ wss 3416   {cpr 3982   <.cop 3986   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Fincfn 7595   CCcc 9563   RRcr 9564   1c1 9566    + caddc 9568    < clt 9701    <_ cle 9702    - cmin 9886   2c2 10687   NN0cn0 10898   ZZcz 10966   #chash 12547   USGrph cusg 25106   Neighbors cnbgra 25194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-xadd 11439  df-fz 11814  df-hash 12548  df-usgra 25109  df-nbgra 25197  df-vdgr 25671
This theorem is referenced by:  nbhashuvtx  25693
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