Theorem uvtx01vtx 29484

Description: If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Revised by AV, 14-Feb-2022.)

Hypotheses

Ref	Expression
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)
isuvtx.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
uvtx01vtx	⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (♯‘𝑉) = 1))

Proof of Theorem uvtx01vtx

Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvtxel.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	uvtxval 29474	. . . 4 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
3	2	a1i 11	. . 3 ⊢ (𝐸 = ∅ → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
4	3	neeq1d 2993	. 2 ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅))
5		rabn0 4317	. . 3 ⊢ ({𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))
6	5	a1i 11	. 2 ⊢ (𝐸 = ∅ → ({𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
7		falseral0 4442	. . . . . . . . . 10 ⊢ ((∀𝑛 ¬ 𝑛 ∈ ∅ ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) → (𝑉 ∖ {𝑣}) = ∅)
8	7	ex 413	. . . . . . . . 9 ⊢ (∀𝑛 ¬ 𝑛 ∈ ∅ → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅))
9		noel 4266	. . . . . . . . 9 ⊢ ¬ 𝑛 ∈ ∅
10	8, 9	mpg 1804	. . . . . . . 8 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅)
11		ssdif0 4294	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 ∖ {𝑣}) = ∅)
12		sssn 4757	. . . . . . . . . 10 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 = ∅ ∨ 𝑉 = {𝑣}))
13		ne0i 4269	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 → 𝑉 ≠ ∅)
14		eqneqall 2945	. . . . . . . . . . . 12 ⊢ (𝑉 = ∅ → (𝑉 ≠ ∅ → 𝑉 = {𝑣}))
15	13, 14	syl5 34	. . . . . . . . . . 11 ⊢ (𝑉 = ∅ → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
16		ax-1 6	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
17	15, 16	jaoi 863	. . . . . . . . . 10 ⊢ ((𝑉 = ∅ ∨ 𝑉 = {𝑣}) → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
18	12, 17	sylbi 218	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑣} → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
19	11, 18	sylbir 236	. . . . . . . 8 ⊢ ((𝑉 ∖ {𝑣}) = ∅ → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
20	10, 19	syl 17	. . . . . . 7 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
21	20	impcom 408	. . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) → 𝑉 = {𝑣})
22		vsnid 4595	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑣}
23		eleq2 2828	. . . . . . . 8 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝑣}))
24	22, 23	mpbiri 259	. . . . . . 7 ⊢ (𝑉 = {𝑣} → 𝑣 ∈ 𝑉)
25		ralel 3056	. . . . . . . 8 ⊢ ∀𝑛 ∈ ∅ 𝑛 ∈ ∅
26		difeq1 4050	. . . . . . . . . 10 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ({𝑣} ∖ {𝑣}))
27		difid 4304	. . . . . . . . . 10 ⊢ ({𝑣} ∖ {𝑣}) = ∅
28	26, 27	eqtrdi 2790	. . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ∅)
29	28	raleqdv 3297	. . . . . . . 8 ⊢ (𝑉 = {𝑣} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ ↔ ∀𝑛 ∈ ∅ 𝑛 ∈ ∅))
30	25, 29	mpbiri 259	. . . . . . 7 ⊢ (𝑉 = {𝑣} → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅)
31	24, 30	jca 516	. . . . . 6 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
32	21, 31	impbii 210	. . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ 𝑉 = {𝑣})
33	32	a1i 11	. . . 4 ⊢ (𝐸 = ∅ → ((𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ 𝑉 = {𝑣}))
34	33	exbidv 1928	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ ∃𝑣 𝑉 = {𝑣}))
35		isuvtx.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
36	35	eqeq1i 2744	. . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
37		nbgr0edg 29444	. . . . . . 7 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
38	36, 37	sylbi 218	. . . . . 6 ⊢ (𝐸 = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
39	38	eleq2d 2825	. . . . 5 ⊢ (𝐸 = ∅ → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ ∅))
40	39	rexralbidv 3205	. . . 4 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
41		df-rex 3064	. . . 4 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
42	40, 41	bitrdi 288	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅)))
43	1	fvexi 6841	. . . 4 ⊢ 𝑉 ∈ V
44		hash1snb 14372	. . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
45	43, 44	mp1i 13	. . 3 ⊢ (𝐸 = ∅ → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
46	34, 42, 45	3bitr4d 312	. 2 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (♯‘𝑉) = 1))
47	4, 6, 46	3bitrd 306	1 ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (♯‘𝑉) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  {crab 3391  Vcvv 3431   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  {csn 4555  ‘cfv 6485  (class class class)co 7356  1c1 11030  ♯chash 14283  Vtxcvtx 29083  Edgcedg 29134   NeighbVtx cnbgr 29419  UnivVtxcuvtx 29472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-hash 14284  df-nbgr 29420  df-uvtx 29473

This theorem is referenced by: (None)