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Theorem usgfisALTlem1 40268
Description: Lemma 1 for usgfisALT 40270: The set of edges is the union of the edges containing a specific vertex and the edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.)
Hypotheses
Ref Expression
usgresvm1ALT.v  |-  V  =  ( 1st `  G
)
usgresvm1ALT.e  |-  E  =  ( Edges  `  G )
usgresvm1ALT.f  |-  F  =  { e  e.  E  |  N  e/  e }
Assertion
Ref Expression
usgfisALTlem1  |-  E  =  ( F  u.  {
e  e.  E  |  N  e.  e }
)
Distinct variable groups:    e, E    e, G    e, N    e, V
Allowed substitution hint:    F( e)

Proof of Theorem usgfisALTlem1
StepHypRef Expression
1 df-nel 2644 . . . . . 6  |-  ( N  e/  e  <->  -.  N  e.  e )
21bicomi 207 . . . . 5  |-  ( -.  N  e.  e  <->  N  e/  e )
32a1i 11 . . . 4  |-  ( e  e.  E  ->  ( -.  N  e.  e  <->  N  e/  e ) )
43rabbiia 3019 . . 3  |-  { e  e.  E  |  -.  N  e.  e }  =  { e  e.  E  |  N  e/  e }
54uneq1i 3575 . 2  |-  ( { e  e.  E  |  -.  N  e.  e }  u.  { e  e.  E  |  N  e.  e } )  =  ( { e  e.  E  |  N  e/  e }  u.  { e  e.  E  |  N  e.  e } )
6 rabxm 3758 . . 3  |-  E  =  ( { e  e.  E  |  N  e.  e }  u.  {
e  e.  E  |  -.  N  e.  e } )
76equncomi 3571 . 2  |-  E  =  ( { e  e.  E  |  -.  N  e.  e }  u.  {
e  e.  E  |  N  e.  e }
)
8 usgresvm1ALT.f . . 3  |-  F  =  { e  e.  E  |  N  e/  e }
98uneq1i 3575 . 2  |-  ( F  u.  { e  e.  E  |  N  e.  e } )  =  ( { e  e.  E  |  N  e/  e }  u.  { e  e.  E  |  N  e.  e } )
105, 7, 93eqtr4i 2503 1  |-  E  =  ( F  u.  {
e  e.  E  |  N  e.  e }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    = wceq 1452    e. wcel 1904    e/ wnel 2642   {crab 2760    u. cun 3388   ` cfv 5589   1stc1st 6810   Edges cedg 25137
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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-nel 2644  df-ral 2761  df-rab 2765  df-v 3033  df-un 3395
This theorem is referenced by:  usgfisALTlem2  40269
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