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Theorem 0conngr 27372
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 4217 . 2 𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓𝑝 𝑓(𝑘(PathsOn‘∅)𝑛)𝑝
2 0ex 4924 . . 3 ∅ ∈ V
3 vtxval0 26152 . . . . 5 (Vtx‘∅) = ∅
43eqcomi 2780 . . . 4 ∅ = (Vtx‘∅)
54isconngr 27369 . . 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 4063   class class class wbr 4786  cfv 6031  (class class class)co 6793  Vtxcvtx 26095  PathsOncpthson 26845  ConnGraphcconngr 27366
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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-slot 16068  df-base 16070  df-vtx 26097  df-conngr 27367
This theorem is referenced by: (None)
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