Theorem predgclnbgrel 48315

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 1149	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → (𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉))
2		simp3 1139	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ∈ 𝐸)
3		sseq2 3948	. . . . 5 ⊢ (𝑒 = {𝑋, 𝑁} → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
4	3	adantl 481	. . . 4 ⊢ (((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) ∧ 𝑒 = {𝑋, 𝑁}) → ({𝑋, 𝑁} ⊆ 𝑒 ↔ {𝑋, 𝑁} ⊆ {𝑋, 𝑁}))
5		ssidd 3945	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → {𝑋, 𝑁} ⊆ {𝑋, 𝑁})
6	2, 4, 5	rspcedvd 3566	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)
7	6	olcd 875	. 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 48304	. 2 ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
11	1, 7, 10	sylanbrc 584	1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3889  {cpr 4569  ‘cfv 6498  (class class class)co 7367  Vtxcvtx 29065  Edgcedg 29116   ClNeighbVtx cclnbgr 48294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-clnbgr 48295

This theorem is referenced by:  grlimgredgex  48476  grlimgrtri  48479