Theorem predgclnbgrel 49210

Description: If a (not necessarily proper) unordered pair containing a vertex is an edge, the other vertex is in the closed neighborhood of the first vertex. (Contributed by AV, 23-Aug-2025.)

Hypotheses

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

Assertion

Ref	Expression
predgclnbgrel	⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))

Proof of Theorem predgclnbgrel

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 1166	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
2		simp3 1156	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ∈ 𝐸)
3		sseq2 3939	. . . . 5 ⊢ (𝑒 = {𝑋, 𝑁} → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
4	3	adantl 487	. . . 4 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) ∧ 𝑒 = {𝑋, 𝑁}) → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
5		ssidd 3936	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ⊆ {𝑋, 𝑁})
6	2, 4, 5	rspcedvd 3582	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)
7	6	olcd 888	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒))
8		predgclnbgrel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
9		predgclnbgrel.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
10	8, 9	clnbgrel 49199	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
11	1, 7, 10	sylanbrc 595	1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  {cpr 4590  ‘cfv 6566  (class class class)co 7451  Vtxcvtx 29631  Edgcedg 29682   ClNeighbVtx cclnbgr 49189

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-clnbgr 49190

This theorem is used by:  grlimgredgex  49371  grlimgrtri  49374