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Theorem usgr0 39482
Description: The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
Assertion
Ref Expression
usgr0  |-  (/)  e. USGraph

Proof of Theorem usgr0
StepHypRef Expression
1 f10 5859 . . 3  |-  (/) : (/) -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x
)  =  2 }
2 dm0 5054 . . . 4  |-  dom  (/)  =  (/)
3 f1eq2 5788 . . . 4  |-  ( dom  (/)  =  (/)  ->  ( (/) : dom  (/) -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x )  =  2 }  <->  (/) : (/) -1-1-> { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 } ) )
42, 3ax-mp 5 . . 3  |-  ( (/) : dom  (/) -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x )  =  2 }  <->  (/) : (/) -1-1-> { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 } )
51, 4mpbir 214 . 2  |-  (/) : dom  (/) -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x )  =  2 }
6 0ex 4528 . . 3  |-  (/)  e.  _V
7 vtxval0 39292 . . . . 5  |-  (Vtx `  (/) )  =  (/)
87eqcomi 2480 . . . 4  |-  (/)  =  (Vtx
`  (/) )
9 iedgval0 39293 . . . . 5  |-  (iEdg `  (/) )  =  (/)
109eqcomi 2480 . . . 4  |-  (/)  =  (iEdg `  (/) )
118, 10isusgr 39401 . . 3  |-  ( (/)  e.  _V  ->  ( (/)  e. USGraph  <->  (/) : dom  (/) -1-1-> { x  e.  ( ~P (/)  \  { (/) } )  |  ( # `  x )  =  2 } ) )
126, 11ax-mp 5 . 2  |-  ( (/)  e. USGraph  <->  (/)
: dom  (/) -1-1-> { x  e.  ( ~P (/)  \  { (/)
} )  |  (
# `  x )  =  2 } )
135, 12mpbir 214 1  |-  (/)  e. USGraph
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031    \ cdif 3387   (/)c0 3722   ~Pcpw 3942   {csn 3959   dom cdm 4839   -1-1->wf1 5586   ` cfv 5589   2c2 10681   #chash 12553  Vtxcvtx 39251  iEdgciedg 39252   USGraph cusgr 39397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fv 5597  df-slot 15203  df-base 15204  df-edgf 39246  df-vtx 39253  df-iedg 39254  df-usgr 39399
This theorem is referenced by:  cusgr0  39658
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