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Theorem 0conngr 27373
 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 4218 . 2 𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝
2 0ex 4925 . . 3 ∅ ∈ V
3 vtxval0 26153 . . . . 5 (Vtx‘∅) = ∅
43eqcomi 2780 . . . 4 ∅ = (Vtx‘∅)
54isconngr 27370 . . 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 221 1 ∅ ∈ ConnGraph
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  Vcvv 3351  ∅c0 4064   class class class wbr 4787  ‘cfv 6032  (class class class)co 6794  Vtxcvtx 26096  PathsOncpthson 26846  ConnGraphcconngr 27367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5995  df-fun 6034  df-fv 6040  df-ov 6797  df-slot 16069  df-base 16071  df-vtx 26098  df-conngr 27368 This theorem is referenced by: (None)
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