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Theorem frgrncvvdeqlem1 41574
Description: Lemma 1 for frgrncvvdeq 41585. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 8-May-2021.)
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 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))

Proof of Theorem frgrncvvdeqlem1
StepHypRef Expression
1 frgrncvvdeq.y . . 3 (𝜑𝑌𝑉)
21adantr 479 . 2 ((𝜑𝑥𝐷) → 𝑌𝑉)
3 frgrncvvdeq.xy . . . . 5 (𝜑𝑌𝐷)
4 df-nel 2687 . . . . . 6 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
5 eleq1a 2587 . . . . . . 7 (𝑥𝐷 → (𝑌 = 𝑥𝑌𝐷))
65con3rr3 149 . . . . . 6 𝑌𝐷 → (𝑥𝐷 → ¬ 𝑌 = 𝑥))
74, 6sylbi 205 . . . . 5 (𝑌𝐷 → (𝑥𝐷 → ¬ 𝑌 = 𝑥))
83, 7syl 17 . . . 4 (𝜑 → (𝑥𝐷 → ¬ 𝑌 = 𝑥))
98imp 443 . . 3 ((𝜑𝑥𝐷) → ¬ 𝑌 = 𝑥)
10 elsng 4042 . . . . 5 (𝑌𝑉 → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
111, 10syl 17 . . . 4 (𝜑 → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
1211adantr 479 . . 3 ((𝜑𝑥𝐷) → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
139, 12mtbird 313 . 2 ((𝜑𝑥𝐷) → ¬ 𝑌 ∈ {𝑥})
142, 13eldifd 3455 1 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wne 2684  wnel 2685  cdif 3441  {csn 4028  {cpr 4030  cmpt 4541  cfv 5689  crio 6387  (class class class)co 6426  Vtxcvtx 40334  Edgcedga 40456   NeighbVtx cnbgr 40655   FriendGraph cfrgr 41533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-nel 2687  df-v 3079  df-dif 3447  df-sn 4029
This theorem is referenced by:  frgrncvvdeqlem3  41576  frgrncvvdeqlem4  41577
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