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Theorem nbgrssvtx 27040
Description: The neighbors of a vertex 𝐾 in a graph form a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)
Hypothesis
Ref Expression
nbgrisvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrssvtx (𝐺 NeighbVtx 𝐾) ⊆ 𝑉

Proof of Theorem nbgrssvtx
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 nbgrisvtx.v . . 3 𝑉 = (Vtx‘𝐺)
21nbgrisvtx 27039 . 2 (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛𝑉)
32ssriv 3974 1 (𝐺 NeighbVtx 𝐾) ⊆ 𝑉
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wss 3939  cfv 6351  (class class class)co 7151  Vtxcvtx 26697   NeighbVtx cnbgr 27030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-nbgr 27031
This theorem is referenced by:  fusgreghash2wspv  28030
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