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Theorem frgrancvvdeqlem1 26557
 Description: Lemma 1 for frgrancvvdeq 26569. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
frgrancvvdeq.ny 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
frgrancvvdeq.x (𝜑𝑋𝑉)
frgrancvvdeq.y (𝜑𝑌𝑉)
frgrancvvdeq.ne (𝜑𝑋𝑌)
frgrancvvdeq.xy (𝜑𝑌𝐷)
frgrancvvdeq.f (𝜑𝑉 FriendGrph 𝐸)
frgrancvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
Assertion
Ref Expression
frgrancvvdeqlem1 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
Distinct variable groups:   𝑦,𝐷   𝑥,𝑦,𝑉   𝑥,𝐸,𝑦   𝑦,𝑌   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐷(𝑥)   𝑁(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrancvvdeqlem1
StepHypRef Expression
1 frgrancvvdeq.y . . 3 (𝜑𝑌𝑉)
21adantr 480 . 2 ((𝜑𝑥𝐷) → 𝑌𝑉)
3 frgrancvvdeq.xy . . . . 5 (𝜑𝑌𝐷)
4 df-nel 2783 . . . . . 6 (𝑌𝐷 ↔ ¬ 𝑌𝐷)
5 eleq1a 2683 . . . . . . 7 (𝑥𝐷 → (𝑌 = 𝑥𝑌𝐷))
65con3rr3 150 . . . . . 6 𝑌𝐷 → (𝑥𝐷 → ¬ 𝑌 = 𝑥))
74, 6sylbi 206 . . . . 5 (𝑌𝐷 → (𝑥𝐷 → ¬ 𝑌 = 𝑥))
83, 7syl 17 . . . 4 (𝜑 → (𝑥𝐷 → ¬ 𝑌 = 𝑥))
98imp 444 . . 3 ((𝜑𝑥𝐷) → ¬ 𝑌 = 𝑥)
10 elsng 4139 . . . . 5 (𝑌𝑉 → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
111, 10syl 17 . . . 4 (𝜑 → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
1211adantr 480 . . 3 ((𝜑𝑥𝐷) → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
139, 12mtbird 314 . 2 ((𝜑𝑥𝐷) → ¬ 𝑌 ∈ {𝑥})
142, 13eldifd 3551 1 ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781   ∖ cdif 3537  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ℩crio 6510  (class class class)co 6549   Neighbors cnbgra 25946   FriendGrph cfrgra 26515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-v 3175  df-dif 3543  df-sn 4126 This theorem is referenced by:  frgrancvvdeqlem3  26559  frgrancvvdeqlem4  26560
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