MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0vconngr Structured version   Visualization version   GIF version

Theorem 0vconngr 27569
Description: A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
Assertion
Ref Expression
0vconngr ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph)

Proof of Theorem 0vconngr
Dummy variables 𝑓 𝑘 𝑛 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rzal 4295 . . 3 ((Vtx‘𝐺) = ∅ → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)
21adantl 475 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)
3 eqid 2825 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
43isconngr 27565 . . 3 (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
54adantr 474 . 2 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
62, 5mpbird 249 1 ((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wex 1880  wcel 2166  wral 3117  c0 4144   class class class wbr 4873  cfv 6123  (class class class)co 6905  Vtxcvtx 26294  PathsOncpthson 27016  ConnGraphcconngr 27562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-conngr 27563
This theorem is referenced by:  1conngr  27570
  Copyright terms: Public domain W3C validator