Theorem vopnbgrelself 49089

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 538	. 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 2760	. . . . . . 7 ⊢ 𝑁 = 𝑁
3	2	jctl 533	. . . . . 6 ⊢ (𝑒 = {𝑁} → (𝑁 = 𝑁 ∧ 𝑒 = {𝑁}))
4	3	olcd 888	. . . . 5 ⊢ (𝑒 = {𝑁} → ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))
5		eqneqall 2966	. . . . . . . 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 1134	. . . . . 6 ⊢ ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) → 𝑒 = {𝑁})
8		simpr 490	. . . . . 6 ⊢ ((𝑁 = 𝑁 ∧ 𝑒 = {𝑁}) → 𝑒 = {𝑁})
9	7, 8	jaoi 871	. . . . 5 ⊢ (((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})) → 𝑒 = {𝑁})
10	4, 9	impbii 212	. . . 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 3186	. 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 49088	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑁 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑁 ∈ 𝑒) ∨ (𝑁 = 𝑁 ∧ 𝑒 = {𝑁})))))
17	1, 12, 16	3bitr4rd 315	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))

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: (None)