Theorem predgclnbgrel 48330

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 1154	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
2		simp3 1144	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ∈ 𝐸)
3		sseq2 3941	. . . . 5 ⊢ (𝑒 = {𝑋, 𝑁} → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
4	3	adantl 482	. . . 4 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) ∧ 𝑒 = {𝑋, 𝑁}) → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
5		ssidd 3938	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ⊆ {𝑋, 𝑁})
6	2, 4, 5	rspcedvd 3562	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)
7	6	olcd 880	. 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 48319	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
11	1, 7, 10	sylanbrc 589	1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   ⊆ wss 3883  {cpr 4557  ‘cfv 6485  (class class class)co 7356  Vtxcvtx 29083  Edgcedg 29134   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-clnbgr 48310

This theorem is referenced by:  grlimgredgex  48491  grlimgrtri  48494