Theorem clnbgrel 48752

Description: Characterization of a member 𝑁 of the closed neighborhood of a vertex 𝑋 in a graph 𝐺. (Contributed by AV, 9-May-2025.)

Hypotheses

Ref	Expression
clnbgrel.v	⊢ 𝑉 = (Vtx‘𝐺)
clnbgrel.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
clnbgrel	⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))

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

Proof of Theorem clnbgrel

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrel.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	clnbgrcl 48745	. . 3 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
3	2	pm4.71ri 570	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)))
4		clnbgrel.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
5	1, 4	clnbgrval 48746	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
6	5	eleq2d 2848	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ 𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒})))
7		elun 4103	. . . . 5 ⊢ (𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 ∈ {𝑋} ∨ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}))
8		elsn2g 4628	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ {𝑋} ↔ 𝑁 = 𝑋))
9		preq2 4698	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → {𝑋, 𝑛} = {𝑋, 𝑁})
10	9	sseq1d 3965	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ({𝑋, 𝑛} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ 𝑒))
11	10	rexbidv 3188	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
12	11	elrab 3648	. . . . . . 7 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
13	12	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒} ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
14	8, 13	orbi12d 932	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 ∈ {𝑋} ∨ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
15	7, 14	bitrid 286	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ ({𝑋} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑋, 𝑛} ⊆ 𝑒}) ↔ (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
16		eleq1 2850	. . . . . . . . 9 ⊢ (𝑁 = 𝑋 → (𝑁 ∈ 𝑉 ↔ 𝑋 ∈ 𝑉))
17	16	biimparc 485	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → 𝑁 ∈ 𝑉)
18		orc 881	. . . . . . . . 9 ⊢ (𝑁 = 𝑋 → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
19	18	adantl 487	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
20	17, 19	jca 521	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 = 𝑋) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
21	20	ex 418	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑁 = 𝑋 → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
22		olc 882	. . . . . . . 8 ⊢ (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
23	22	anim2i 629	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
24	23	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
25	21, 24	jaod 873	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) → (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
26		orc 881	. . . . . . . 8 ⊢ (𝑁 = 𝑋 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
27	26	a1i 11	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑁 = 𝑋 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
28		olc 882	. . . . . . . . 9 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
29	28	ex 418	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
30	29	adantl 487	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒 → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
31	27, 30	jaod 873	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → ((𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
32	31	expimpd 459	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) → (𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
33	25, 32	impbid 215	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑁 = 𝑋 ∨ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
34	6, 15, 33	3bitrd 308	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
35	34	pm5.32i 585	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋)) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
36		anass 474	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) ↔ (𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))))
37	36	bicomi 227	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))) ↔ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
38		ancom 466	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) ↔ (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
39	37, 38	bianbi 639	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
40	3, 35, 39	3bitri 300	1 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  {crab 3414   ∪ cun 3900   ⊆ wss 3902  {csn 4587  {cpr 4589  ‘cfv 6537  (class class class)co 7417  Vtxcvtx 29461  Edgcedg 29512   ClNeighbVtx cclnbgr 48742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-clnbgr 48743

This theorem is used by:  clnbgrvtxel  48753  clnbgrisvtx  48754  clnbgrsym  48762  predgclnbgrel  48763  clnbgredg  48764  clnbgrgrimlem  48857  clnbgrgrim  48858