Theorem nbgrisvtx 29432

Description: Every neighbor 𝑁 of a vertex 𝐾 is a vertex. (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
nbgrisvtx	⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁 ∈ 𝑉)

Proof of Theorem nbgrisvtx

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrisvtx.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2737	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	nbgrel 29431	. 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))
4		simp1l 1199	. 2 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉) ∧ 𝑁 ≠ 𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁 ∈ 𝑉)
5	3, 4	sylbi 217	1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁 ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ⊆ wss 3903  {cpr 4584  ‘cfv 6502  (class class class)co 7370  Vtxcvtx 29087  Edgcedg 29138   NeighbVtx cnbgr 29423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-nbgr 29424

This theorem is referenced by:  nbgrssvtx  29433  nbgrnself2  29451  nbgrssovtx  29452  frgrnbnb  30386  frgrncvvdeqlem2  30393  frgrncvvdeqlem3  30394  frgrncvvdeqlem9  30400  numclwwlk1lem2foa  30447  numclwwlk1lem2fo  30451