Theorem dfclnbgr6 48347

Description: Alternate definition of the closed neighborhood of a vertex as union of the vertex with its semiopen neighborhood. (Contributed by AV, 17-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
dfclnbgr6	⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑈))

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

Allowed substitution hints:   𝑈(𝑒,𝑛)

Proof of Theorem dfclnbgr6

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 873	. . . . . . 7 ⊢ (𝑣 = 𝑁 → (𝑣 = 𝑁 ∨ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))))
2	1	a1d 25	. . . . . 6 ⊢ (𝑣 = 𝑁 → (𝑁 ∈ 𝑉 → (𝑣 = 𝑁 ∨ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))))))
3		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → 𝑣 ≠ 𝑁)
4	3	anim1i 621	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) → (𝑣 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
5		3anass 1100	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ↔ (𝑣 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
6	4, 5	sylibr 235	. . . . . . . . . . . . 13 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) → (𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))
7	6	orcd 879	. . . . . . . . . . . 12 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) → ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))
8	7	ex 413	. . . . . . . . . . 11 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → ((𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) → ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
9		3simpc 1156	. . . . . . . . . . . . 13 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
11		vsnid 4595	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ {𝑣}
12		eleq2 2828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = {𝑣} → (𝑣 ∈ 𝑒 ↔ 𝑣 ∈ {𝑣}))
13	11, 12	mpbiri 259	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑣} → 𝑣 ∈ 𝑒)
14	13	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑁 ∧ 𝑒 = {𝑣}) → 𝑣 ∈ 𝑒)
15		eleq1 2827	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑁 → (𝑣 ∈ 𝑒 ↔ 𝑁 ∈ 𝑒))
16	15	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑁 ∧ 𝑒 = {𝑣}) → (𝑣 ∈ 𝑒 ↔ 𝑁 ∈ 𝑒))
17	14, 16	mpbid 233	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑁 ∧ 𝑒 = {𝑣}) → 𝑁 ∈ 𝑒)
18	17, 14	jca 516	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑁 ∧ 𝑒 = {𝑣}) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → ((𝑣 = 𝑁 ∧ 𝑒 = {𝑣}) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
20	10, 19	jaod 865	. . . . . . . . . . 11 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → (((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
21	8, 20	impbid 213	. . . . . . . . . 10 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → ((𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ↔ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
22	21	rexbidv 3163	. . . . . . . . 9 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ↔ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
23	22	anbi2d 636	. . . . . . . 8 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))))
24	23	olcd 880	. . . . . . 7 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑉) → (𝑣 = 𝑁 ∨ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))))
25	24	ex 413	. . . . . 6 ⊢ (𝑣 ≠ 𝑁 → (𝑁 ∈ 𝑉 → (𝑣 = 𝑁 ∨ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))))))
26	2, 25	pm2.61ine 3017	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑣 = 𝑁 ∨ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))))
27		orbidi 960	. . . . 5 ⊢ ((𝑣 = 𝑁 ∨ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))) ↔ ((𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))) ↔ (𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))))
28	26, 27	sylib 219	. . . 4 ⊢ (𝑁 ∈ 𝑉 → ((𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))) ↔ (𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))))
29		elun 4083	. . . . 5 ⊢ (𝑣 ∈ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) ↔ (𝑣 ∈ {𝑁} ∨ 𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
30		velsn 4571	. . . . . 6 ⊢ (𝑣 ∈ {𝑁} ↔ 𝑣 = 𝑁)
31		eleq1 2827	. . . . . . . . 9 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ 𝑒 ↔ 𝑣 ∈ 𝑒))
32	31	anbi2d 636	. . . . . . . 8 ⊢ (𝑛 = 𝑣 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
33	32	rexbidv 3163	. . . . . . 7 ⊢ (𝑛 = 𝑣 → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
34	33	elrab 3629	. . . . . 6 ⊢ (𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
35	30, 34	orbi12i 920	. . . . 5 ⊢ ((𝑣 ∈ {𝑁} ∨ 𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) ↔ (𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))))
36	29, 35	bitri 276	. . . 4 ⊢ (𝑣 ∈ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) ↔ (𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))))
37		elun 4083	. . . . 5 ⊢ (𝑣 ∈ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}) ↔ (𝑣 ∈ {𝑁} ∨ 𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}))
38		neeq1 2996	. . . . . . . . . 10 ⊢ (𝑛 = 𝑣 → (𝑛 ≠ 𝑁 ↔ 𝑣 ≠ 𝑁))
39	38, 31	3anbi13d 1446	. . . . . . . . 9 ⊢ (𝑛 = 𝑣 → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
40		eqeq1 2743	. . . . . . . . . 10 ⊢ (𝑛 = 𝑣 → (𝑛 = 𝑁 ↔ 𝑣 = 𝑁))
41		sneq 4565	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑣 → {𝑛} = {𝑣})
42	41	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑛 = 𝑣 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑣}))
43	40, 42	anbi12d 638	. . . . . . . . 9 ⊢ (𝑛 = 𝑣 → ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) ↔ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))
44	39, 43	orbi12d 924	. . . . . . . 8 ⊢ (𝑛 = 𝑣 → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
45	44	rexbidv 3163	. . . . . . 7 ⊢ (𝑛 = 𝑣 → (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
46	45	elrab 3629	. . . . . 6 ⊢ (𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
47	30, 46	orbi12i 920	. . . . 5 ⊢ ((𝑣 ∈ {𝑁} ∨ 𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}) ↔ (𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))))
48	37, 47	bitri 276	. . . 4 ⊢ (𝑣 ∈ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}) ↔ (𝑣 = 𝑁 ∨ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))))
49	28, 36, 48	3bitr4g 315	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑣 ∈ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) ↔ 𝑣 ∈ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})))
50	49	eqrdv 2737	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}))
51		dfvopnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
52		dfvopnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
53	51, 52	dfclnbgr2 48314	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
54		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
55	51, 52, 54	dfvopnbgr2 48344	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
56	55	uneq2d 4098	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ 𝑈) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))}))
57	50, 53, 56	3eqtr4d 2784	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑈))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391   ∪ cun 3881  {csn 4555  ‘cfv 6485  (class class class)co 7356  Vtxcvtx 29083  Edgcedg 29134   NeighbVtx cnbgr 29419   ClNeighbVtx cclnbgr 48309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-nbgr 29420  df-clnbgr 48310

This theorem is referenced by: (None)