Theorem vopnbgrelself 47839

Description: A vertex 𝑁 is a member of its semiopen neighborhood iff there is a loop joining the vertex with itself. (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
vopnbgrelself	⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))

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

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

Proof of Theorem vopnbgrelself

Step	Hyp	Ref	Expression
1		ibar 528	. 2 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})) ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))))
2		eqid 2729	. . . . . . 7 ⊢ 𝑁 = 𝑁
3	2	jctl 523	. . . . . 6 ⊢ (𝑒 = {𝑁} → (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))
4	3	olcd 874	. . . . 5 ⊢ (𝑒 = {𝑁} → ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))
5		eqneqall 2936	. . . . . . . 8 ⊢ (𝑁 = 𝑁 → (𝑁 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁})))
6	2, 5	ax-mp 5	. . . . . . 7 ⊢ (𝑁 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁}))
7	6	3impib 1116	. . . . . 6 ⊢ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁})
8		simpr 484	. . . . . 6 ⊢ ((𝑁 = 𝑁 ∧ 𝑒 = {𝑁}) → 𝑒 = {𝑁})
9	7, 8	jaoi 857	. . . . 5 ⊢ (((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})) → 𝑒 = {𝑁})
10	4, 9	impbii 209	. . . 4 ⊢ (𝑒 = {𝑁} ↔ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))
11	10	a1i 11	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑒 = {𝑁} ↔ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))))
12	11	rexbidv 3153	. 2 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 𝑒 = {𝑁} ↔ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))))
13		dfvopnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
14		dfvopnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
15		dfvopnbgr2.u	. . 3 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
16	13, 14, 15	vopnbgrel 47838	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))))
17	1, 12, 16	3bitr4rd 312	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3394  {csn 4577  ‘cfv 6482  (class class class)co 7349  Vtxcvtx 28941  Edgcedg 28992   NeighbVtx cnbgr 29277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-nbgr 29278

This theorem is referenced by: (None)