Theorem vopnbgrel 49088

Description: Characterization of a member 𝑋 of the semiopen neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)

Hypotheses

Ref	Expression
dfvopnbgr2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfvopnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)
dfvopnbgr2.u	⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}

Assertion

Ref	Expression
vopnbgrel	⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑛,𝐸   𝑒,𝑋,𝑛

Allowed substitution hints:   𝑈(𝑒, 𝑛)   𝐺(𝑛)

Proof of Theorem vopnbgrel

Step	Hyp	Ref	Expression
1		dfvopnbgr2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		dfvopnbgr2.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
4	1, 2, 3	dfvopnbgr2 49087	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
5	4	eleq2d 2846	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}))
6		neeq1 3017	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ≠ 𝑁 ↔ 𝑋 ≠ 𝑁))
7		eleq1 2848	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝑒 ↔ 𝑋 ∈ 𝑒))
8	6, 7	3anbi13d 1466	. . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒)))
9		eqeq1 2764	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 = 𝑁 ↔ 𝑋 = 𝑁))
10		sneq 4590	. . . . . . 7 ⊢ (𝑛 = 𝑋 → {𝑛} = {𝑋})
11	10	eqeq2d 2771	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑋}))
12	9, 11	anbi12d 644	. . . . 5 ⊢ (𝑛 = 𝑋 → ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) ↔ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))
13	8, 12	orbi12d 932	. . . 4 ⊢ (𝑛 = 𝑋 → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))
14	13	rexbidv 3186	. . 3 ⊢ (𝑛 = 𝑋 → (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))
15	14	elrab 3644	. 2 ⊢ (𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋}))))
16	5, 15	bitrdi 290	1 ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086  {crab 3412  {csn 4584  ‘cfv 6543  (class class class)co 7424  Vtxcvtx 29583  Edgcedg 29634   NeighbVtx cnbgr 29922

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-nbgr 29923

This theorem is used by:  vopnbgrelself  49089