Theorem clnbgrssedg 48908

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 48907	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑣 ∈ 𝐾)) → 𝑣 ∈ 𝑁)
4	3	3exp2 1373	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐾 ∈ 𝐸 → (𝑋 ∈ 𝐾 → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁))))
5	4	3imp 1128	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → (𝑣 ∈ 𝐾 → 𝑣 ∈ 𝑁))
6	5	ssrdv 3937	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ‘cfv 6537  (class class class)co 7418  Edgcedg 29618  UHGraphcuhgr 29627   ClNeighbVtx cclnbgr 48885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-edg 29619  df-uhgr 29629  df-clnbgr 48886

This theorem is used by:  clnbgrvtxedg  49061