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	⊢ (𝜑 → 𝑋 ∈ 𝑉)
frgrancvvdeq.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
frgrancvvdeq.ne	⊢ (𝜑 → 𝑋 ≠ 𝑌)
frgrancvvdeq.xy	⊢ (𝜑 → 𝑌 ∉ 𝐷)
frgrancvvdeq.f	⊢ (𝜑 → 𝑉 FriendGrph 𝐸)
frgrancvvdeq.a	⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ ran 𝐸))

Assertion

Ref	Expression
frgrancvvdeqlem1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 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

Step	Hyp	Ref	Expression
1		frgrancvvdeq.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
2	1	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑌 ∈ 𝑉)
3		frgrancvvdeq.xy	. . . . 5 ⊢ (𝜑 → 𝑌 ∉ 𝐷)
4		df-nel 2783	. . . . . 6 ⊢ (𝑌 ∉ 𝐷 ↔ ¬ 𝑌 ∈ 𝐷)
5		eleq1a 2683	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (𝑌 = 𝑥 → 𝑌 ∈ 𝐷))
6	5	con3rr3 150	. . . . . 6 ⊢ (¬ 𝑌 ∈ 𝐷 → (𝑥 ∈ 𝐷 → ¬ 𝑌 = 𝑥))
7	4, 6	sylbi 206	. . . . 5 ⊢ (𝑌 ∉ 𝐷 → (𝑥 ∈ 𝐷 → ¬ 𝑌 = 𝑥))
8	3, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐷 → ¬ 𝑌 = 𝑥))
9	8	imp 444	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ¬ 𝑌 = 𝑥)
10		elsng 4139	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
11	1, 10	syl 17	. . . 4 ⊢ (𝜑 → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
12	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑌 ∈ {𝑥} ↔ 𝑌 = 𝑥))
13	9, 12	mtbird 314	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ¬ 𝑌 ∈ {𝑥})
14	2, 13	eldifd 3551	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))

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