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Theorem nbhashuvtx1 25488
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 2628 . . 3  |-  ( M  e/  ( <. V ,  E >. Neighbors  N )  <->  -.  M  e.  ( <. V ,  E >. Neighbors  N ) )
4 nbgrassvwo2 25011 . . . . . . 7  |-  ( ( V USGrph  E  /\  M  e/  ( <. V ,  E >. Neighbors  N ) )  -> 
( <. V ,  E >. Neighbors  N )  C_  ( V  \  { M ,  N } ) )
54ex 435 . . . . . 6  |-  ( V USGrph  E  ->  ( M  e/  ( <. V ,  E >. Neighbors  N )  ->  ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { M ,  N } ) ) )
653ad2ant1 1026 . . . . 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 4573 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  { M ,  N } )  e.  _V )
873ad2ant2 1027 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( V  \  { M ,  N } )  e.  _V )
9 hashss 12583 . . . . . . 7  |-  ( ( ( V  \  { M ,  N }
)  e.  _V  /\  ( <. V ,  E >. Neighbors  N )  C_  ( V  \  { M ,  N } ) )  -> 
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  ( # `
 ( V  \  { M ,  N }
) ) )
109ex 435 . . . . . 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 1009 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  V  e.  Fin )
13 simpl 458 . . . . . . . . . . . . 13  |-  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  V )
14 simp3 1007 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  N  e.  V )
15 prssi 4159 . . . . . . . . . . . . 13  |-  ( ( M  e.  V  /\  N  e.  V )  ->  { M ,  N }  C_  V )
1613, 14, 15syl2anr 480 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  { M ,  N }  C_  V )
17 hashssdif 12584 . . . . . . . . . . . 12  |-  ( ( V  e.  Fin  /\  { M ,  N }  C_  V )  ->  ( # `
 ( V  \  { M ,  N }
) )  =  ( ( # `  V
)  -  ( # `  { M ,  N } ) ) )
1812, 16, 17syl2anc 665 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( V  \  { M ,  N } ) )  =  ( ( # `  V
)  -  ( # `  { M ,  N } ) ) )
19 simprr 764 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  M  =/=  N
)
20 hashprg 12569 . . . . . . . . . . . . . 14  |-  ( ( M  e.  V  /\  N  e.  V )  ->  ( M  =/=  N  <->  (
# `  { M ,  N } )  =  2 ) )
2113, 14, 20syl2anr 480 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( M  =/= 
N  <->  ( # `  { M ,  N }
)  =  2 ) )
2219, 21mpbid 213 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  { M ,  N }
)  =  2 )
2322oveq2d 6321 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  V )  -  ( # `
 { M ,  N } ) )  =  ( ( # `  V
)  -  2 ) )
2418, 23eqtrd 2470 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( V  \  { M ,  N } ) )  =  ( ( # `  V
)  -  2 ) )
2524breq2d 4438 . . . . . . . . 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 25487 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e. 
NN0 )
2726nn0zd 11038 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e.  ZZ )
28 hashcl 12535 . . . . . . . . . . . . . . 15  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
29 nn0z 10960 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  e.  NN0  ->  ( # `  V
)  e.  ZZ )
30 peano2zm 10980 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  e.  ZZ  ->  ( ( # `
 V )  - 
1 )  e.  ZZ )
3128, 29, 303syl 18 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  -  1 )  e.  ZZ )
32313ad2ant2 1027 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  V )  -  1 )  e.  ZZ )
33 zltlem1 10989 . . . . . . . . . . . . 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 665 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( ( # `  V
)  -  1 )  -  1 ) ) )
3528nn0cnd 10927 . . . . . . . . . . . . . . . 16  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  CC )
36353ad2ant2 1027 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 V )  e.  CC )
37 1cnd 9658 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  1  e.  CC )
3836, 37, 37subsub4d 10016 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( ( # `  V
)  -  1 )  -  1 )  =  ( ( # `  V
)  -  ( 1  +  1 ) ) )
39 1p1e2 10723 . . . . . . . . . . . . . . 15  |-  ( 1  +  1 )  =  2
4039oveq2i 6316 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  -  ( 1  +  1 ) )  =  ( ( # `  V
)  -  2 )
4138, 40syl6eq 2486 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( ( # `  V
)  -  1 )  -  1 )  =  ( ( # `  V
)  -  2 ) )
4241breq2d 4438 . . . . . . . . . . . 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 256 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( # `  V )  -  2 ) ) )
4443adantr 466 . . . . . . . . . 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 10926 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e.  RR )
4645adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( <. V ,  E >. Neighbors  N
) )  e.  RR )
4746adantr 466 . . . . . . . . . . . . 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 462 . . . . . . . . . . . . 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 9770 . . . . . . . . . . . 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 435 . . . . . . . . . . 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 2638 . . . . . . . . . . . 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 238 . . . . . . . . 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 218 . . . . . . . 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 435 . . . . . . 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 216 . 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 167 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625    e/ wnel 2626   _Vcvv 3087    \ cdif 3439    C_ wss 3442   {cpr 4004   <.cop 4008   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Fincfn 7577   CCcc 9536   RRcr 9537   1c1 9539    + caddc 9541    < clt 9674    <_ cle 9675    - cmin 9859   2c2 10659   NN0cn0 10869   ZZcz 10937   #chash 12512   USGrph cusg 24903   Neighbors cnbgra 24990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-xadd 11410  df-fz 11783  df-hash 12513  df-usgra 24906  df-nbgra 24993  df-vdgr 25467
This theorem is referenced by:  nbhashuvtx  25489
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