Theorem uvtxnbgrb 29751

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 29750	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
3		simpl 487	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ 𝑉)
4		raleleq 3335	. . . . . 6 ⊢ ((𝑉 ∖ {𝑁}) = (𝐺 NeighbVtx 𝑁) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
5	4	eqcoms 2771	. . . . 5 ⊢ ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
6	5	adantl 486	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
7	1	uvtxel 29738	. . . 4 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
8	3, 6, 7	sylanbrc 594	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})) → 𝑁 ∈ (UnivVtx‘𝐺))
9	8	ex 417	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}) → 𝑁 ∈ (UnivVtx‘𝐺)))
10	2, 9	impbid2 229	1 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∖ cdif 3902  {csn 4589  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29346   NeighbVtx cnbgr 29682  UnivVtxcuvtx 29735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-nbgr 29683  df-uvtx 29736

This theorem is referenced by:  nbusgrvtxm1uvtx  29755  uvtxupgrres  29758  nbcplgr  29784