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Theorem uhgrvd0nedgb 39595
Description: A vertex in a hypergraph has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.)
Hypotheses
Ref Expression
vtxdushgrfvedg.v  |-  V  =  (Vtx `  G )
vtxdushgrfvedg.e  |-  E  =  (Edg `  G )
vtxdushgrfvedg.d  |-  D  =  (VtxDeg `  G )
Assertion
Ref Expression
uhgrvd0nedgb  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
( D `  U
)  =  0  <->  -.  E. i  e.  dom  (iEdg `  G ) U  e.  ( (iEdg `  G
) `  i )
) )
Distinct variable groups:    i, E    i, G    U, i    i, V
Allowed substitution hint:    D( i)

Proof of Theorem uhgrvd0nedgb
StepHypRef Expression
1 vtxdushgrfvedg.d . . . . 5  |-  D  =  (VtxDeg `  G )
21fveq1i 5889 . . . 4  |-  ( D `
 U )  =  ( (VtxDeg `  G
) `  U )
3 vtxdushgrfvedg.v . . . . 5  |-  V  =  (Vtx `  G )
4 eqid 2462 . . . . 5  |-  (iEdg `  G )  =  (iEdg `  G )
5 eqid 2462 . . . . 5  |-  dom  (iEdg `  G )  =  dom  (iEdg `  G )
63, 4, 5vtxdgval 39579 . . . 4  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
(VtxDeg `  G ) `  U )  =  ( ( # `  {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } ) +e
( # `  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } ) ) )
72, 6syl5eq 2508 . . 3  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  ( D `  U )  =  ( ( # `  { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G ) `  i
) } ) +e ( # `  {
i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } ) ) )
87eqeq1d 2464 . 2  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
( D `  U
)  =  0  <->  (
( # `  { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } ) +e
( # `  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } ) )  =  0 ) )
9 fvex 5898 . . . . . . 7  |-  (iEdg `  G )  e.  _V
109dmex 6753 . . . . . 6  |-  dom  (iEdg `  G )  e.  _V
1110rabex 4568 . . . . 5  |-  { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) }  e.  _V
12 hashxnn0 39136 . . . . 5  |-  ( { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) }  e.  _V  ->  (
# `  { i  e.  dom  (iEdg `  G
)  |  U  e.  ( (iEdg `  G
) `  i ) } )  e. NN0* )
1311, 12ax-mp 5 . . . 4  |-  ( # `  { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G ) `  i
) } )  e. NN0*
1410rabex 4568 . . . . 5  |-  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } }  e.  _V
15 hashxnn0 39136 . . . . 5  |-  ( { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } }  e.  _V  ->  ( # `  {
i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } )  e. NN0*
)
1614, 15ax-mp 5 . . . 4  |-  ( # `  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G
) `  i )  =  { U } }
)  e. NN0*
1713, 16pm3.2i 461 . . 3  |-  ( (
# `  { i  e.  dom  (iEdg `  G
)  |  U  e.  ( (iEdg `  G
) `  i ) } )  e. NN0*  /\  ( # `  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } )  e. NN0*
)
18 xnn0xadd0 39134 . . 3  |-  ( ( ( # `  {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } )  e. NN0*  /\  ( # `  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } )  e. NN0*
)  ->  ( (
( # `  { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } ) +e
( # `  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } ) )  =  0  <->  ( ( # `
 { i  e. 
dom  (iEdg `  G )  |  U  e.  (
(iEdg `  G ) `  i ) } )  =  0  /\  ( # `
 { i  e. 
dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i
)  =  { U } } )  =  0 ) ) )
1917, 18mp1i 13 . 2  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
( ( # `  {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } ) +e
( # `  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } ) )  =  0  <->  ( ( # `
 { i  e. 
dom  (iEdg `  G )  |  U  e.  (
(iEdg `  G ) `  i ) } )  =  0  /\  ( # `
 { i  e. 
dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i
)  =  { U } } )  =  0 ) ) )
20 hasheq0 12576 . . . . . 6  |-  ( { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) }  e.  _V  ->  ( ( # `  {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } )  =  0  <->  { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G ) `  i
) }  =  (/) ) )
2111, 20ax-mp 5 . . . . 5  |-  ( (
# `  { i  e.  dom  (iEdg `  G
)  |  U  e.  ( (iEdg `  G
) `  i ) } )  =  0  <->  { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G ) `  i
) }  =  (/) )
22 hasheq0 12576 . . . . . 6  |-  ( { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } }  e.  _V  ->  ( ( # `  {
i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } )  =  0  <->  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G
) `  i )  =  { U } }  =  (/) ) )
2314, 22ax-mp 5 . . . . 5  |-  ( (
# `  { i  e.  dom  (iEdg `  G
)  |  ( (iEdg `  G ) `  i
)  =  { U } } )  =  0  <->  { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G
) `  i )  =  { U } }  =  (/) )
2421, 23anbi12i 708 . . . 4  |-  ( ( ( # `  {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } )  =  0  /\  ( # `  {
i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } )  =  0 )  <->  ( {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) }  =  (/)  /\  {
i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } }  =  (/) ) )
25 rabeq0 3766 . . . . 5  |-  ( { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) }  =  (/)  <->  A. i  e.  dom  (iEdg `  G
)  -.  U  e.  ( (iEdg `  G
) `  i )
)
26 rabeq0 3766 . . . . 5  |-  ( { i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } }  =  (/)  <->  A. i  e.  dom  (iEdg `  G )  -.  (
(iEdg `  G ) `  i )  =  { U } )
2725, 26anbi12i 708 . . . 4  |-  ( ( { i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G ) `  i
) }  =  (/)  /\ 
{ i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G
) `  i )  =  { U } }  =  (/) )  <->  ( A. i  e.  dom  (iEdg `  G )  -.  U  e.  ( (iEdg `  G
) `  i )  /\  A. i  e.  dom  (iEdg `  G )  -.  ( (iEdg `  G
) `  i )  =  { U } ) )
28 ralnex 2846 . . . . . . 7  |-  ( A. i  e.  dom  (iEdg `  G )  -.  ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } )  <->  -.  E. i  e.  dom  (iEdg `  G
) ( U  e.  ( (iEdg `  G
) `  i )  \/  ( (iEdg `  G
) `  i )  =  { U } ) )
2928bicomi 207 . . . . . 6  |-  ( -. 
E. i  e.  dom  (iEdg `  G ) ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } )  <->  A. i  e.  dom  (iEdg `  G
)  -.  ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } ) )
30 ioran 497 . . . . . . 7  |-  ( -.  ( U  e.  ( (iEdg `  G ) `  i )  \/  (
(iEdg `  G ) `  i )  =  { U } )  <->  ( -.  U  e.  ( (iEdg `  G ) `  i
)  /\  -.  (
(iEdg `  G ) `  i )  =  { U } ) )
3130ralbii 2831 . . . . . 6  |-  ( A. i  e.  dom  (iEdg `  G )  -.  ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } )  <->  A. i  e.  dom  (iEdg `  G
) ( -.  U  e.  ( (iEdg `  G
) `  i )  /\  -.  ( (iEdg `  G ) `  i
)  =  { U } ) )
32 r19.26 2929 . . . . . 6  |-  ( A. i  e.  dom  (iEdg `  G ) ( -.  U  e.  ( (iEdg `  G ) `  i
)  /\  -.  (
(iEdg `  G ) `  i )  =  { U } )  <->  ( A. i  e.  dom  (iEdg `  G )  -.  U  e.  ( (iEdg `  G
) `  i )  /\  A. i  e.  dom  (iEdg `  G )  -.  ( (iEdg `  G
) `  i )  =  { U } ) )
3329, 31, 323bitri 279 . . . . 5  |-  ( -. 
E. i  e.  dom  (iEdg `  G ) ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } )  <->  ( A. i  e.  dom  (iEdg `  G )  -.  U  e.  ( (iEdg `  G
) `  i )  /\  A. i  e.  dom  (iEdg `  G )  -.  ( (iEdg `  G
) `  i )  =  { U } ) )
3433bicomi 207 . . . 4  |-  ( ( A. i  e.  dom  (iEdg `  G )  -.  U  e.  ( (iEdg `  G ) `  i
)  /\  A. i  e.  dom  (iEdg `  G
)  -.  ( (iEdg `  G ) `  i
)  =  { U } )  <->  -.  E. i  e.  dom  (iEdg `  G
) ( U  e.  ( (iEdg `  G
) `  i )  \/  ( (iEdg `  G
) `  i )  =  { U } ) )
3524, 27, 343bitri 279 . . 3  |-  ( ( ( # `  {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } )  =  0  /\  ( # `  {
i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } )  =  0 )  <->  -.  E. i  e.  dom  (iEdg `  G
) ( U  e.  ( (iEdg `  G
) `  i )  \/  ( (iEdg `  G
) `  i )  =  { U } ) )
36 snidg 4006 . . . . . . . . 9  |-  ( U  e.  V  ->  U  e.  { U } )
37 eleq2 2529 . . . . . . . . 9  |-  ( ( (iEdg `  G ) `  i )  =  { U }  ->  ( U  e.  ( (iEdg `  G ) `  i
)  <->  U  e.  { U } ) )
3836, 37syl5ibrcom 230 . . . . . . . 8  |-  ( U  e.  V  ->  (
( (iEdg `  G
) `  i )  =  { U }  ->  U  e.  ( (iEdg `  G ) `  i
) ) )
3938adantl 472 . . . . . . 7  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
( (iEdg `  G
) `  i )  =  { U }  ->  U  e.  ( (iEdg `  G ) `  i
) ) )
40 pm4.72 892 . . . . . . 7  |-  ( ( ( (iEdg `  G
) `  i )  =  { U }  ->  U  e.  ( (iEdg `  G ) `  i
) )  <->  ( U  e.  ( (iEdg `  G
) `  i )  <->  ( ( (iEdg `  G
) `  i )  =  { U }  \/  U  e.  ( (iEdg `  G ) `  i
) ) ) )
4139, 40sylib 201 . . . . . 6  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  ( U  e.  ( (iEdg `  G ) `  i
)  <->  ( ( (iEdg `  G ) `  i
)  =  { U }  \/  U  e.  ( (iEdg `  G ) `  i ) ) ) )
42 orcom 393 . . . . . 6  |-  ( ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } )  <->  ( (
(iEdg `  G ) `  i )  =  { U }  \/  U  e.  ( (iEdg `  G
) `  i )
) )
4341, 42syl6rbbr 272 . . . . 5  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
( U  e.  ( (iEdg `  G ) `  i )  \/  (
(iEdg `  G ) `  i )  =  { U } )  <->  U  e.  ( (iEdg `  G ) `  i ) ) )
4443rexbidv 2913 . . . 4  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  ( E. i  e.  dom  (iEdg `  G ) ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } )  <->  E. i  e.  dom  (iEdg `  G
) U  e.  ( (iEdg `  G ) `  i ) ) )
4544notbid 300 . . 3  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  ( -.  E. i  e.  dom  (iEdg `  G ) ( U  e.  ( (iEdg `  G ) `  i
)  \/  ( (iEdg `  G ) `  i
)  =  { U } )  <->  -.  E. i  e.  dom  (iEdg `  G
) U  e.  ( (iEdg `  G ) `  i ) ) )
4635, 45syl5bb 265 . 2  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
( ( # `  {
i  e.  dom  (iEdg `  G )  |  U  e.  ( (iEdg `  G
) `  i ) } )  =  0  /\  ( # `  {
i  e.  dom  (iEdg `  G )  |  ( (iEdg `  G ) `  i )  =  { U } } )  =  0 )  <->  -.  E. i  e.  dom  (iEdg `  G
) U  e.  ( (iEdg `  G ) `  i ) ) )
478, 19, 463bitrd 287 1  |-  ( ( G  e. UHGraph  /\  U  e.  V )  ->  (
( D `  U
)  =  0  <->  -.  E. i  e.  dom  (iEdg `  G ) U  e.  ( (iEdg `  G
) `  i )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   {crab 2753   _Vcvv 3057   (/)c0 3743   {csn 3980   dom cdm 4853   ` cfv 5601  (class class class)co 6315   0cc0 9565   +ecxad 11436   #chash 12547  NN0*cxnn0 39116  Vtxcvtx 39147  iEdgciedg 39148   UHGraph cuhgr 39193  Edgcedga 39258  VtxDegcvtxdg 39576
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-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-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  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-n0 10899  df-z 10967  df-uz 11189  df-xadd 11439  df-fz 11814  df-hash 12548  df-xnn0 39117  df-vtxdg 39577
This theorem is referenced by:  vtxduhgr0nedg  39596  vtxduhgr0edgnel  39598
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