MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbhashuvtx1 Structured version   Unicode version

Theorem nbhashuvtx1 24738
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 . . . 4  |-  ( M  e.  ( <. V ,  E >. Neighbors  N )  ->  (
( M  e.  V  /\  M  =/=  N
)  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) )
21a1d 25 . . 3  |-  ( M  e.  ( <. V ,  E >. Neighbors  N )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) ) )
32a1d 25 . 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 ) ) ) ) )
4 df-nel 2665 . . 3  |-  ( M  e/  ( <. V ,  E >. Neighbors  N )  <->  -.  M  e.  ( <. V ,  E >. Neighbors  N ) )
5 nbgrassvwo2 24261 . . . . . . 7  |-  ( ( V USGrph  E  /\  M  e/  ( <. V ,  E >. Neighbors  N ) )  -> 
( <. V ,  E >. Neighbors  N )  C_  ( V  \  { M ,  N } ) )
65ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( M  e/  ( <. V ,  E >. Neighbors  N )  ->  ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { M ,  N } ) ) )
763ad2ant1 1017 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( M  e/  ( <. V ,  E >. Neighbors  N )  ->  ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { M ,  N } ) ) )
8 difexg 4601 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  { M ,  N } )  e.  _V )
983ad2ant2 1018 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( V  \  { M ,  N } )  e.  _V )
10 hashss 12454 . . . . . . 7  |-  ( ( ( V  \  { M ,  N }
)  e.  _V  /\  ( <. V ,  E >. Neighbors  N )  C_  ( V  \  { M ,  N } ) )  -> 
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  ( # `
 ( V  \  { M ,  N }
) ) )
1110ex 434 . . . . . 6  |-  ( ( V  \  { M ,  N } )  e. 
_V  ->  ( ( <. V ,  E >. Neighbors  N
)  C_  ( V  \  { M ,  N } )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  <_ 
( # `  ( V 
\  { M ,  N } ) ) ) )
129, 11syl 16 . . . . 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 } ) ) ) )
13 simpl2 1000 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  V  e.  Fin )
14 simpl 457 . . . . . . . . . . . . 13  |-  ( ( M  e.  V  /\  M  =/=  N )  ->  M  e.  V )
15 simp3 998 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  N  e.  V )
16 prssi 4189 . . . . . . . . . . . . 13  |-  ( ( M  e.  V  /\  N  e.  V )  ->  { M ,  N }  C_  V )
1714, 15, 16syl2anr 478 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  { M ,  N }  C_  V )
18 hashssdif 12455 . . . . . . . . . . . 12  |-  ( ( V  e.  Fin  /\  { M ,  N }  C_  V )  ->  ( # `
 ( V  \  { M ,  N }
) )  =  ( ( # `  V
)  -  ( # `  { M ,  N } ) ) )
1913, 17, 18syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( V  \  { M ,  N } ) )  =  ( ( # `  V
)  -  ( # `  { M ,  N } ) ) )
20 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  M  =/=  N
)
21 hashprg 12440 . . . . . . . . . . . . . 14  |-  ( ( M  e.  V  /\  N  e.  V )  ->  ( M  =/=  N  <->  (
# `  { M ,  N } )  =  2 ) )
2214, 15, 21syl2anr 478 . . . . . . . . . . . . 13  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( M  =/= 
N  <->  ( # `  { M ,  N }
)  =  2 ) )
2320, 22mpbid 210 . . . . . . . . . . . 12  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  { M ,  N }
)  =  2 )
2423oveq2d 6311 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( ( # `  V )  -  ( # `
 { M ,  N } ) )  =  ( ( # `  V
)  -  2 ) )
2519, 24eqtrd 2508 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( V  \  { M ,  N } ) )  =  ( ( # `  V
)  -  2 ) )
2625breq2d 4465 . . . . . . . . 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 ) ) )
27 nbhashnn0 24737 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e. 
NN0 )
28 nn0z 10899 . . . . . . . . . . . . . 14  |-  ( (
# `  ( <. V ,  E >. Neighbors  N ) )  e.  NN0  ->  (
# `  ( <. V ,  E >. Neighbors  N ) )  e.  ZZ )
2927, 28syl 16 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e.  ZZ )
30 hashcl 12408 . . . . . . . . . . . . . . 15  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
31 nn0z 10899 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  e.  NN0  ->  ( # `  V
)  e.  ZZ )
32 peano2zm 10918 . . . . . . . . . . . . . . 15  |-  ( (
# `  V )  e.  ZZ  ->  ( ( # `
 V )  - 
1 )  e.  ZZ )
3330, 31, 323syl 20 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  -  1 )  e.  ZZ )
34333ad2ant2 1018 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  V )  -  1 )  e.  ZZ )
35 zltlem1 10927 . . . . . . . . . . . . 13  |-  ( ( ( # `  ( <. V ,  E >. Neighbors  N
) )  e.  ZZ  /\  ( ( # `  V
)  -  1 )  e.  ZZ )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( ( # `  V
)  -  1 )  -  1 ) ) )
3629, 34, 35syl2anc 661 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( ( # `  V
)  -  1 )  -  1 ) ) )
3730nn0cnd 10866 . . . . . . . . . . . . . . . 16  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  CC )
38373ad2ant2 1018 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 V )  e.  CC )
39 ax-1cn 9562 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
4039a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  1  e.  CC )
4138, 40, 40subsub4d 9973 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( ( # `  V
)  -  1 )  -  1 )  =  ( ( # `  V
)  -  ( 1  +  1 ) ) )
42 1p1e2 10661 . . . . . . . . . . . . . . 15  |-  ( 1  +  1 )  =  2
4342oveq2i 6306 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  -  ( 1  +  1 ) )  =  ( ( # `  V
)  -  2 )
4441, 43syl6eq 2524 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( ( # `  V
)  -  1 )  -  1 )  =  ( ( # `  V
)  -  2 ) )
4544breq2d 4465 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( ( # `  V
)  -  1 )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( # `  V )  -  2 ) ) )
4636, 45bitrd 253 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  (
( # `  ( <. V ,  E >. Neighbors  N
) )  <  (
( # `  V )  -  1 )  <->  ( # `  ( <. V ,  E >. Neighbors  N
) )  <_  (
( # `  V )  -  2 ) ) )
4746adantr 465 . . . . . . . . . 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 ) ) )
4827nn0red 10865 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  N ) )  e.  RR )
4948adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  /\  ( M  e.  V  /\  M  =/=  N ) )  ->  ( # `  ( <. V ,  E >. Neighbors  N
) )  e.  RR )
5049adantr 465 . . . . . . . . . . . . 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 )
51 simpr 461 . . . . . . . . . . . . 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 ) )
5250, 51ltned 9732 . . . . . . . . . . . 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 ) )
5352ex 434 . . . . . . . . . . 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 ) ) )
54 eqneqall 2674 . . . . . . . . . . . 12  |-  ( (
# `  ( <. V ,  E >. Neighbors  N ) )  =  ( (
# `  V )  -  1 )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  =/=  (
( # `  V )  -  1 )  ->  M  e.  ( <. V ,  E >. Neighbors  N ) ) )
5554com12 31 . . . . . . . . . . 11  |-  ( (
# `  ( <. V ,  E >. Neighbors  N ) )  =/=  ( (
# `  V )  -  1 )  -> 
( ( # `  ( <. V ,  E >. Neighbors  N
) )  =  ( ( # `  V
)  -  1 )  ->  M  e.  (
<. V ,  E >. Neighbors  N
) ) )
5653, 55syl6 33 . . . . . . . . . 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
) ) ) )
5747, 56sylbird 235 . . . . . . . . 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
) ) ) )
5826, 57sylbid 215 . . . . . . . 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
) ) ) )
5958ex 434 . . . . . . 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
) ) ) ) )
6059com23 78 . . . . . 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
) ) ) ) )
6160com34 83 . . . . 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 ) ) ) ) )
627, 12, 613syld 55 . . . 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 ) ) ) ) )
6362com12 31 . . 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 ) ) ) ) )
644, 63sylbir 213 . 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 ) ) ) ) )
653, 64pm2.61i 164 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    e/ wnel 2663   _Vcvv 3118    \ cdif 3478    C_ wss 3481   {cpr 4035   <.cop 4039   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   RRcr 9503   1c1 9505    + caddc 9507    < clt 9640    <_ cle 9641    - cmin 9817   2c2 10597   NN0cn0 10807   ZZcz 10876   #chash 12385   USGrph cusg 24153   Neighbors cnbgra 24240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-xadd 11331  df-fz 11685  df-hash 12386  df-usgra 24156  df-nbgra 24243  df-vdgr 24717
This theorem is referenced by:  nbhashuvtx  24739
  Copyright terms: Public domain W3C validator