Theorem clnbgrssedg 47800

Description: The vertices connected by an edge are a subset of the neighborhood of each of these vertices. (Contributed by AV, 26-May-2025.) (Proof shortened by AV, 24-Aug-2025.)

Hypotheses

Ref	Expression
clnbgrssedg.e	⊢ 𝐸 = (Edg‘𝐺)
clnbgrssedg.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)

Assertion

Ref	Expression
clnbgrssedg	⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)

Proof of Theorem clnbgrssedg

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgrssedg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
2		clnbgrssedg.n	. . . . 5 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)
3	1, 2	clnbgredg 47799	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝑁)
4	3	3exp2 1354	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ 𝐸 → (𝑋 ∈ 𝐾 → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁))))
5	4	3imp 1110	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁))
6	5	ssrdv 3969	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931  ‘cfv 6541  (class class class)co 7413  Edgcedg 28993  UHGraphcuhgr 29002   ClNeighbVtx cclnbgr 47778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-edg 28994  df-uhgr 29004  df-clnbgr 47779

This theorem is referenced by:  grlimgrtrilem1  47934