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Theorem nbgrael 24291
 Description: The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.)
Assertion
Ref Expression
nbgrael Neighbors

Proof of Theorem nbgrael
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 24285 . . . 4 Neighbors
21mpt2xopn0yelv 6939 . . 3 Neighbors
32pm4.71rd 635 . 2 Neighbors Neighbors
4 nbgraop 24288 . . . . . 6 Neighbors
54eleq2d 2511 . . . . 5 Neighbors
6 preq2 4091 . . . . . . 7
76eleq1d 2510 . . . . . 6
87elrab 3241 . . . . 5
95, 8syl6bb 261 . . . 4 Neighbors
109pm5.32da 641 . . 3 Neighbors
11 3anass 976 . . 3
1210, 11syl6bbr 263 . 2 Neighbors
133, 12bitrd 253 1 Neighbors
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 972   wceq 1381   wcel 1802  crab 2795  cpr 4012  cop 4016   crn 4986  cfv 5574  (class class class)co 6277  c1st 6779  c2nd 6780   Neighbors cnbgra 24282 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-nbgra 24285 This theorem is referenced by:  nbgrasym  24304  nbgraf1olem1  24306  usg2spot2nb  24930  extwwlkfablem1  24939
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