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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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