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Theorem 0conngr 41359
 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
0conngr ∅ ∈ ConnGraph

Proof of Theorem 0conngr
Dummy variables 𝑓 𝑘 𝑛 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 4028 . 2 𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝
2 0ex 4718 . . 3 ∅ ∈ V
3 vtxval0 25714 . . . . 5 (Vtx‘∅) = ∅
43eqcomi 2619 . . . 4 ∅ = (Vtx‘∅)
54isconngr 41356 . . 3 (∅ ∈ V → (∅ ∈ ConnGraph ↔ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝))
62, 5ax-mp 5 . 2 (∅ ∈ ConnGraph ↔ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝)
71, 6mpbir 220 1 ∅ ∈ ConnGraph
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  PathsOncpthson 40921  ConnGraphcconngr 41353 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-slot 15699  df-base 15700  df-vtx 25675  df-conngr 41354 This theorem is referenced by: (None)
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