Theorem dfnbgr6 47843

Description: Alternate definition of the (open) neighborhood of a vertex as a difference of its semiopen neighborhood and the singleton of itself. (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
dfnbgr6	⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))

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 dfnbgr6

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabdif 4321	. . 3 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}
2		3anass 1095	. . . . . . . . . . . . 13 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ↔ (𝑣 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
3	2	biimpri 228	. . . . . . . . . . . 12 ⊢ ((𝑣 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) → (𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))
4	3	orcd 874	. . . . . . . . . . 11 ⊢ ((𝑣 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) → ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))
5	4	ex 412	. . . . . . . . . 10 ⊢ (𝑣 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) → ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
6		3simpc 1151	. . . . . . . . . . . 12 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))
7	6	a1i 11	. . . . . . . . . . 11 ⊢ (𝑣 ≠ 𝑁 → ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
8		eqneqall 2951	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑁 → (𝑣 ≠ 𝑁 → (𝑒 = {𝑣} → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))))
9	8	com12 32	. . . . . . . . . . . 12 ⊢ (𝑣 ≠ 𝑁 → (𝑣 = 𝑁 → (𝑒 = {𝑣} → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒))))
10	9	impd 410	. . . . . . . . . . 11 ⊢ (𝑣 ≠ 𝑁 → ((𝑣 = 𝑁 ∧ 𝑒 = {𝑣}) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
11	7, 10	jaod 860	. . . . . . . . . 10 ⊢ (𝑣 ≠ 𝑁 → (((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})) → (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
12	5, 11	impbid 212	. . . . . . . . 9 ⊢ (𝑣 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ↔ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
13	12	rexbidv 3179	. . . . . . . 8 ⊢ (𝑣 ≠ 𝑁 → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ↔ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
14	13	anbi2d 630	. . . . . . 7 ⊢ (𝑣 ≠ 𝑁 → ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))))
15	14	pm5.32ri 575	. . . . . 6 ⊢ (((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ∧ 𝑣 ≠ 𝑁) ↔ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))) ∧ 𝑣 ≠ 𝑁))
16	15	a1i 11	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ∧ 𝑣 ≠ 𝑁) ↔ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))) ∧ 𝑣 ≠ 𝑁)))
17		eldif 3961	. . . . . 6 ⊢ (𝑣 ∈ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) ↔ (𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∧ ¬ 𝑣 ∈ {𝑁}))
18		elequ1 2115	. . . . . . . . . 10 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ 𝑒 ↔ 𝑣 ∈ 𝑒))
19	18	anbi2d 630	. . . . . . . . 9 ⊢ (𝑛 = 𝑣 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
20	19	rexbidv 3179	. . . . . . . 8 ⊢ (𝑛 = 𝑣 → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
21	20	elrab 3692	. . . . . . 7 ⊢ (𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ↔ (𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
22		velsn 4642	. . . . . . . 8 ⊢ (𝑣 ∈ {𝑁} ↔ 𝑣 = 𝑁)
23	22	necon3bbii 2988	. . . . . . 7 ⊢ (¬ 𝑣 ∈ {𝑁} ↔ 𝑣 ≠ 𝑁)
24	21, 23	anbi12i 628	. . . . . 6 ⊢ ((𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∧ ¬ 𝑣 ∈ {𝑁}) ↔ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ∧ 𝑣 ≠ 𝑁))
25	17, 24	bitri 275	. . . . 5 ⊢ (𝑣 ∈ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) ↔ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)) ∧ 𝑣 ≠ 𝑁))
26		eldif 3961	. . . . . 6 ⊢ (𝑣 ∈ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∖ {𝑁}) ↔ (𝑣 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∧ ¬ 𝑣 ∈ {𝑁}))
27		neeq1 3003	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑣 → (𝑛 ≠ 𝑁 ↔ 𝑣 ≠ 𝑁))
28	27, 18	3anbi13d 1440	. . . . . . . . . 10 ⊢ (𝑛 = 𝑣 → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒)))
29		eqeq1 2741	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑣 → (𝑛 = 𝑁 ↔ 𝑣 = 𝑁))
30		sneq 4636	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑣 → {𝑛} = {𝑣})
31	30	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑣 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑣}))
32	29, 31	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑛 = 𝑣 → ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) ↔ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣})))
33	28, 32	orbi12d 919	. . . . . . . . 9 ⊢ (𝑛 = 𝑣 → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ↔ ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))))
34	33	rexbidv 3179	. . . . . . . 8 ⊢ (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}))))
35	34	elrab 3692	. . . . . . 7 ⊢ (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}))))
36	35, 23	anbi12i 628	. . . . . 6 ⊢ ((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}))) ∧ 𝑣 ≠ 𝑁))
37	26, 36	bitri 275	. . . . 5 ⊢ (𝑣 ∈ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∖ {𝑁}) ↔ ((𝑣 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑣 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑣 ∈ 𝑒) ∨ (𝑣 = 𝑁 ∧ 𝑒 = {𝑣}))) ∧ 𝑣 ≠ 𝑁))
38	16, 25, 37	3bitr4g 314	. . . 4 ⊢ (𝑁 ∈ 𝑉 → (𝑣 ∈ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) ↔ 𝑣 ∈ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∖ {𝑁})))
39	38	eqrdv 2735	. . 3 ⊢ (𝑁 ∈ 𝑉 → ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∖ {𝑁}))
40	1, 39	eqtr3id 2791	. 2 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∖ {𝑁}))
41		dfvopnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
42		dfvopnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
43	41, 42	dfnbgr2 29354	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
44		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
45	41, 42, 44	dfvopnbgr2 47839	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
46	45	difeq1d 4125	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑈 ∖ {𝑁}) = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∖ {𝑁}))
47	40, 43, 46	3eqtr4d 2787	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  {crab 3436   ∖ cdif 3948  {csn 4626  ‘cfv 6561  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-nbgr 29350

This theorem is referenced by:  dfnbgrss2  47845