Theorem clnbgrssedg 47845

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 47844	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝑁)
4	3	3exp2 1355	. . 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 3943	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3905  ‘cfv 6486  (class class class)co 7353  Edgcedg 29011  UHGraphcuhgr 29020   ClNeighbVtx cclnbgr 47822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-edg 29012  df-uhgr 29022  df-clnbgr 47823

This theorem is referenced by:  clnbgrvtxedg  47998