Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  usgfisALTlem1 Structured version   Unicode version

Theorem usgfisALTlem1 32286
Description: Lemma 1 for usgfisALT 32288: 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 2641 . . . . . 6  |-  ( N  e/  e  <->  -.  N  e.  e )
21bicomi 202 . . . . 5  |-  ( -.  N  e.  e  <->  N  e/  e )
32a1i 11 . . . 4  |-  ( e  e.  E  ->  ( -.  N  e.  e  <->  N  e/  e ) )
43rabbiia 3084 . . 3  |-  { e  e.  E  |  -.  N  e.  e }  =  { e  e.  E  |  N  e/  e }
54uneq1i 3639 . 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 3794 . . 3  |-  E  =  ( { e  e.  E  |  N  e.  e }  u.  {
e  e.  E  |  -.  N  e.  e } )
76equncomi 3635 . 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 3639 . 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 2482 1  |-  E  =  ( F  u.  {
e  e.  E  |  N  e.  e }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1383    e. wcel 1804    e/ wnel 2639   {crab 2797    u. cun 3459   ` cfv 5578   1stc1st 6783   Edges cedg 24203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-nel 2641  df-ral 2798  df-rab 2802  df-v 3097  df-un 3466
This theorem is referenced by:  usgfisALTlem2  32287
  Copyright terms: Public domain W3C validator