Theorem clnbgrssedg 47703

Description: The vertices connected by an edge are a subset of the neigborhood 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 47702	. . . 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 1111	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁))
6	5	ssrdv 4014	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ‘cfv 6568  (class class class)co 7443  Edgcedg 29074  UHGraphcuhgr 29083   ClNeighbVtx cclnbgr 47682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7764

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8024  df-2nd 8025  df-edg 29075  df-uhgr 29085  df-clnbgr 47683

This theorem is referenced by:  grlimgrtrilem1  47808