Theorem uvtxnbgrvtx 29591

Description: A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)

Hypothesis

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxnbgrvtx	⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑁   𝑣,𝑉

Proof of Theorem uvtxnbgrvtx

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxel.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	vtxnbuvtx 29589	. . 3 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
3		eleq1w 2845	. . . . . . 7 ⊢ (𝑛 = 𝑣 → (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑣 ∈ (𝐺 NeighbVtx 𝑁)))
4	3	rspcva 3579	. . . . . 6 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑣 ∈ (𝐺 NeighbVtx 𝑁))
5		nbgrsym 29561	. . . . . . 7 ⊢ (𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∈ (𝐺 NeighbVtx 𝑣))
6	5	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝐺 NeighbVtx 𝑁) ↔ 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))
7	4, 6	syl5ibcom 247	. . . . 5 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)) → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))
8	7	expcom 417	. . . 4 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))))
9	8	com23 86	. . 3 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁) → (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣))))
10	2, 9	mpcom 38	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ∈ (𝐺 NeighbVtx 𝑣)))
11	10	ralrimiv 3153	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∖ cdif 3901  {csn 4582  ‘cfv 6521  (class class class)co 7396  Vtxcvtx 29194   NeighbVtx cnbgr 29530  UnivVtxcuvtx 29583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-nbgr 29531  df-uvtx 29584

This theorem is referenced by: (None)