Theorem dfnbgrss2 48776

Description: Subset chain for different kinds of neighborhoods of a vertex. (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}))}
dfsclnbgr6.s	⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}

Assertion

Ref	Expression
dfnbgrss2	⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))

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

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

Proof of Theorem dfnbgrss2

Step	Hyp	Ref	Expression
1		dfvopnbgr2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		dfvopnbgr2.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
4	1, 2, 3	dfnbgr6 48774	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
5		difss 4083	. . 3 ⊢ (𝑈 ∖ {𝑁}) ⊆ 𝑈
6	4, 5	eqsstrdi 3975	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑈)
7		ssun1 4124	. . 3 ⊢ 𝑈 ⊆ (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒})
8		dfsclnbgr6.s	. . . 4 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
9	1, 2, 3, 8	dfsclnbgr6 48775	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
10	7, 9	sseqtrrid 3974	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑈 ⊆ 𝑆)
11	1, 8, 2	dfnbgrss 48769	. . 3 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
12	11	simprd 501	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))
13	6, 10, 12	3jca 1146	1 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {csn 4584  {cpr 4586  ‘cfv 6533  (class class class)co 7414  Vtxcvtx 29454  Edgcedg 29505   NeighbVtx cnbgr 29793   ClNeighbVtx cclnbgr 48735

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 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

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 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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-nbgr 29794  df-clnbgr 48736

This theorem is used by: (None)