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Theorem isusgra0 24020
Description: The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
Assertion
Ref Expression
isusgra0  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
Distinct variable groups:    x, E    x, V    x, W    x, X

Proof of Theorem isusgra0
StepHypRef Expression
1 isusgra 24017 . 2  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
2 vex 3116 . . . . . . . . . . . . . 14  |-  x  e. 
_V
3 hasheq0 12395 . . . . . . . . . . . . . 14  |-  ( x  e.  _V  ->  (
( # `  x )  =  0  <->  x  =  (/) ) )
42, 3ax-mp 5 . . . . . . . . . . . . 13  |-  ( (
# `  x )  =  0  <->  x  =  (/) )
5 2ne0 10624 . . . . . . . . . . . . . . . . 17  |-  2  =/=  0
65a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  0  ->  2  =/=  0 )
7 id 22 . . . . . . . . . . . . . . . 16  |-  ( (
# `  x )  =  0  ->  ( # `
 x )  =  0 )
86, 7neeqtrrd 2767 . . . . . . . . . . . . . . 15  |-  ( (
# `  x )  =  0  ->  2  =/=  ( # `  x
) )
98necomd 2738 . . . . . . . . . . . . . 14  |-  ( (
# `  x )  =  0  ->  ( # `
 x )  =/=  2 )
109neneqd 2669 . . . . . . . . . . . . 13  |-  ( (
# `  x )  =  0  ->  -.  ( # `  x )  =  2 )
114, 10sylbir 213 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  -.  ( # `
 x )  =  2 )
1211con2i 120 . . . . . . . . . . 11  |-  ( (
# `  x )  =  2  ->  -.  x  =  (/) )
1312adantl 466 . . . . . . . . . 10  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  -.  x  =  (/) )
14 elsn 4041 . . . . . . . . . 10  |-  ( x  e.  { (/) }  <->  x  =  (/) )
1513, 14sylnibr 305 . . . . . . . . 9  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  -.  x  e.  {
(/) } )
1615biantrud 507 . . . . . . . 8  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  ( x  e. 
~P V  <->  ( x  e.  ~P V  /\  -.  x  e.  { (/) } ) ) )
17 eldif 3486 . . . . . . . 8  |-  ( x  e.  ( ~P V  \  { (/) } )  <->  ( x  e.  ~P V  /\  -.  x  e.  { (/) } ) )
1816, 17syl6rbbr 264 . . . . . . 7  |-  ( ( ( V  e.  W  /\  E  e.  X
)  /\  ( # `  x
)  =  2 )  ->  ( x  e.  ( ~P V  \  { (/) } )  <->  x  e.  ~P V ) )
1918ex 434 . . . . . 6  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( ( # `  x
)  =  2  -> 
( x  e.  ( ~P V  \  { (/)
} )  <->  x  e.  ~P V ) ) )
2019pm5.32rd 640 . . . . 5  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( ( x  e.  ( ~P V  \  { (/) } )  /\  ( # `  x )  =  2 )  <->  ( x  e.  ~P V  /\  ( # `
 x )  =  2 ) ) )
2120abbidv 2603 . . . 4  |-  ( ( V  e.  W  /\  E  e.  X )  ->  { x  |  ( x  e.  ( ~P V  \  { (/) } )  /\  ( # `  x )  =  2 ) }  =  {
x  |  ( x  e.  ~P V  /\  ( # `  x )  =  2 ) } )
22 df-rab 2823 . . . 4  |-  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x
)  =  2 }  =  { x  |  ( x  e.  ( ~P V  \  { (/)
} )  /\  ( # `
 x )  =  2 ) }
23 df-rab 2823 . . . 4  |-  { x  e.  ~P V  |  (
# `  x )  =  2 }  =  { x  |  (
x  e.  ~P V  /\  ( # `  x
)  =  2 ) }
2421, 22, 233eqtr4g 2533 . . 3  |-  ( ( V  e.  W  /\  E  e.  X )  ->  { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 }  =  { x  e.  ~P V  |  ( # `  x
)  =  2 } )
25 f1eq3 5776 . . 3  |-  ( { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  =  { x  e.  ~P V  |  (
# `  x )  =  2 }  ->  ( E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
2624, 25syl 16 . 2  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 }  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
271, 26bitrd 253 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x
)  =  2 } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   {crab 2818   _Vcvv 3113    \ cdif 3473   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   dom cdm 4999   -1-1->wf1 5583   ` cfv 5586   0cc0 9488   2c2 10581   #chash 12367   USGrph cusg 24003
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12368  df-usgra 24006
This theorem is referenced by:  usgraf0  24021  ausisusgra  24028  usgra1  24046  usgraexmpl  24074  usgrares1  24083  cusgraexilem2  24140  isfusgra0  31894  usgresvm1  31912  usgresvm1ALT  31916
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