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Theorem frgrncvvdeqlem1 27279
Description: Lemma 1 for frgrncvvdeq 27289. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlem1 (𝜑𝑋𝑁)

Proof of Theorem frgrncvvdeqlem1
StepHypRef Expression
1 frgrncvvdeq.xy . . . 4 (𝜑𝑌𝐷)
2 df-nel 2927 . . . . 5 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
3 frgrncvvdeq.nx . . . . . 6 𝐷 = (𝐺 NeighbVtx 𝑋)
43eleq2i 2722 . . . . 5 (𝑌𝐷𝑌 ∈ (𝐺 NeighbVtx 𝑋))
52, 4xchbinx 323 . . . 4 (𝑌𝐷 ↔ ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
61, 5sylib 208 . . 3 (𝜑 → ¬ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
7 nbgrsym 26308 . . 3 (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))
86, 7sylnibr 318 . 2 (𝜑 → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
9 frgrncvvdeq.ny . . . 4 𝑁 = (𝐺 NeighbVtx 𝑌)
10 neleq2 2932 . . . 4 (𝑁 = (𝐺 NeighbVtx 𝑌) → (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌)))
119, 10ax-mp 5 . . 3 (𝑋𝑁𝑋 ∉ (𝐺 NeighbVtx 𝑌))
12 df-nel 2927 . . 3 (𝑋 ∉ (𝐺 NeighbVtx 𝑌) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
1311, 12bitri 264 . 2 (𝑋𝑁 ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑌))
148, 13sylibr 224 1 (𝜑𝑋𝑁)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1523  wcel 2030  wne 2823  wnel 2926  {cpr 4212  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984   NeighbVtx cnbgr 26269   FriendGraph cfrgr 27236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-nbgr 26270
This theorem is referenced by:  frgrncvvdeqlem7  27285  frgrncvvdeqlem8  27286  frgrncvvdeqlem9  27287
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