Theorem uvtx01vtx 28692

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 28682	. . . 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 3000	. 2 ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} ≠ ∅))
5		rabn0 4385	. . 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 4519	. . . . . . . . . 10 ⊢ ((∀𝑛 ¬ 𝑛 ∈ ∅ ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) → (𝑉 ∖ {𝑣}) = ∅)
8	7	ex 413	. . . . . . . . 9 ⊢ (∀𝑛 ¬ 𝑛 ∈ ∅ → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅))
9		noel 4330	. . . . . . . . 9 ⊢ ¬ 𝑛 ∈ ∅
10	8, 9	mpg 1799	. . . . . . . 8 ⊢ (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ → (𝑉 ∖ {𝑣}) = ∅)
11		ssdif0 4363	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 ∖ {𝑣}) = ∅)
12		sssn 4829	. . . . . . . . . 10 ⊢ (𝑉 ⊆ {𝑣} ↔ (𝑉 = ∅ ∨ 𝑉 = {𝑣}))
13		ne0i 4334	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ 𝑉 → 𝑉 ≠ ∅)
14		eqneqall 2951	. . . . . . . . . . . 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 855	. . . . . . . . . 10 ⊢ ((𝑉 = ∅ ∨ 𝑉 = {𝑣}) → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
18	12, 17	sylbi 216	. . . . . . . . 9 ⊢ (𝑉 ⊆ {𝑣} → (𝑣 ∈ 𝑉 → 𝑉 = {𝑣}))
19	11, 18	sylbir 234	. . . . . . . 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 4665	. . . . . . . 8 ⊢ 𝑣 ∈ {𝑣}
23		eleq2 2822	. . . . . . . 8 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ {𝑣}))
24	22, 23	mpbiri 257	. . . . . . 7 ⊢ (𝑉 = {𝑣} → 𝑣 ∈ 𝑉)
25		ralel 3064	. . . . . . . 8 ⊢ ∀𝑛 ∈ ∅ 𝑛 ∈ ∅
26		difeq1 4115	. . . . . . . . . 10 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ({𝑣} ∖ {𝑣}))
27		difid 4370	. . . . . . . . . 10 ⊢ ({𝑣} ∖ {𝑣}) = ∅
28	26, 27	eqtrdi 2788	. . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (𝑉 ∖ {𝑣}) = ∅)
29	28	raleqdv 3325	. . . . . . . 8 ⊢ (𝑉 = {𝑣} → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ ↔ ∀𝑛 ∈ ∅ 𝑛 ∈ ∅))
30	25, 29	mpbiri 257	. . . . . . 7 ⊢ (𝑉 = {𝑣} → ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅)
31	24, 30	jca 512	. . . . . 6 ⊢ (𝑉 = {𝑣} → (𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
32	21, 31	impbii 208	. . . . 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 1924	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅) ↔ ∃𝑣 𝑉 = {𝑣}))
35		isuvtx.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
36	35	eqeq1i 2737	. . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅)
37		nbgr0edg 28652	. . . . . . 7 ⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
38	36, 37	sylbi 216	. . . . . 6 ⊢ (𝐸 = ∅ → (𝐺 NeighbVtx 𝑣) = ∅)
39	38	eleq2d 2819	. . . . 5 ⊢ (𝐸 = ∅ → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ ∅))
40	39	rexralbidv 3220	. . . 4 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
41		df-rex 3071	. . . 4 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅))
42	40, 41	bitrdi 286	. . 3 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∃𝑣(𝑣 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ ∅)))
43	1	fvexi 6905	. . . 4 ⊢ 𝑉 ∈ V
44		hash1snb 14381	. . . 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 310	. 2 ⊢ (𝐸 = ∅ → (∃𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (♯‘𝑉) = 1))
47	4, 6, 46	3bitrd 304	1 ⊢ (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (♯‘𝑉) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  {csn 4628  ‘cfv 6543  (class class class)co 7411  1c1 11113  ♯chash 14292  Vtxcvtx 28294  Edgcedg 28345   NeighbVtx cnbgr 28627  UnivVtxcuvtx 28680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-hash 14293  df-nbgr 28628  df-uvtx 28681

This theorem is referenced by: (None)