Theorem uvtxnbgrb 29476

Description: A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 14-Feb-2022.)

Hypothesis

Ref	Expression
uvtxnbgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxnbgrb	⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Proof of Theorem uvtxnbgrb

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxnbgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	uvtxnbgr 29475	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
3		simpl 482	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ 𝑉)
4		raleleq 3312	. . . . . 6 ⊢ ((𝑉 ∖ {𝑁}) = (𝐺 NeighbVtx 𝑁) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
5	4	eqcoms 2744	. . . . 5 ⊢ ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
6	5	adantl 481	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
7	1	uvtxel 29463	. . . 4 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
8	3, 6, 7	sylanbrc 583	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ (UnivVtx‘𝐺))
9	8	ex 412	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → 𝑁 ∈ (UnivVtx‘𝐺)))
10	2, 9	impbid2 226	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∖ cdif 3898  {csn 4580  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29071   NeighbVtx cnbgr 29407  UnivVtxcuvtx 29460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-nbgr 29408  df-uvtx 29461

This theorem is referenced by:  nbusgrvtxm1uvtx  29480  uvtxupgrres  29483  nbcplgr  29509