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Theorem 0vconngr 29443
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 ((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) β†’ 𝐺 ∈ ConnGraph)

Proof of Theorem 0vconngr
Dummy variables 𝑓 π‘˜ 𝑛 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rzal 4508 . . 3 ((Vtxβ€˜πΊ) = βˆ… β†’ βˆ€π‘˜ ∈ (Vtxβ€˜πΊ)βˆ€π‘› ∈ (Vtxβ€˜πΊ)βˆƒπ‘“βˆƒπ‘ 𝑓(π‘˜(PathsOnβ€˜πΊ)𝑛)𝑝)
21adantl 482 . 2 ((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) β†’ βˆ€π‘˜ ∈ (Vtxβ€˜πΊ)βˆ€π‘› ∈ (Vtxβ€˜πΊ)βˆƒπ‘“βˆƒπ‘ 𝑓(π‘˜(PathsOnβ€˜πΊ)𝑛)𝑝)
3 eqid 2732 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43isconngr 29439 . . 3 (𝐺 ∈ π‘Š β†’ (𝐺 ∈ ConnGraph ↔ βˆ€π‘˜ ∈ (Vtxβ€˜πΊ)βˆ€π‘› ∈ (Vtxβ€˜πΊ)βˆƒπ‘“βˆƒπ‘ 𝑓(π‘˜(PathsOnβ€˜πΊ)𝑛)𝑝))
54adantr 481 . 2 ((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) β†’ (𝐺 ∈ ConnGraph ↔ βˆ€π‘˜ ∈ (Vtxβ€˜πΊ)βˆ€π‘› ∈ (Vtxβ€˜πΊ)βˆƒπ‘“βˆƒπ‘ 𝑓(π‘˜(PathsOnβ€˜πΊ)𝑛)𝑝))
62, 5mpbird 256 1 ((𝐺 ∈ π‘Š ∧ (Vtxβ€˜πΊ) = βˆ…) β†’ 𝐺 ∈ ConnGraph)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  βˆ…c0 4322   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  Vtxcvtx 28253  PathsOncpthson 28968  ConnGraphcconngr 29436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-conngr 29437
This theorem is referenced by:  1conngr  29444
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