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Theorem nbgrasym 25012
Description: A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgrasym  |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  K  e.  ( <. V ,  E >. Neighbors  N ) ) )

Proof of Theorem nbgrasym
StepHypRef Expression
1 usgrav 24911 . 2  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 3ancoma 989 . . . . 5  |-  ( ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { K ,  N }  e.  ran  E ) )
3 prcom 4081 . . . . . . 7  |-  { K ,  N }  =  { N ,  K }
43eleq1i 2506 . . . . . 6  |-  ( { K ,  N }  e.  ran  E  <->  { N ,  K }  e.  ran  E )
543anbi3i 1198 . . . . 5  |-  ( ( N  e.  V  /\  K  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) )
62, 5bitri 252 . . . 4  |-  ( ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) )
76a1i 11 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) ) )
8 nbgrael 24999 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  (
<. V ,  E >. Neighbors  K
)  <->  ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E ) ) )
9 nbgrael 24999 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( K  e.  (
<. V ,  E >. Neighbors  N
)  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) ) )
107, 8, 93bitr4d 288 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  (
<. V ,  E >. Neighbors  K
)  <->  K  e.  ( <. V ,  E >. Neighbors  N
) ) )
111, 10syl 17 1  |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  K  e.  ( <. V ,  E >. Neighbors  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1870   _Vcvv 3087   {cpr 4004   <.cop 4008   class class class wbr 4426   ran crn 4855  (class class class)co 6305   USGrph cusg 24903   Neighbors cnbgra 24990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-usgra 24906  df-nbgra 24993
This theorem is referenced by:  uvtxnbgravtx  25068
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