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Theorem 0vconngr 29713
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 4507 . . 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 480 . 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 2730 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43isconngr 29709 . . 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 479 . 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 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  βˆ€wral 3059  βˆ…c0 4321   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Vtxcvtx 28523  PathsOncpthson 29238  ConnGraphcconngr 29706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-conngr 29707
This theorem is referenced by:  1conngr  29714
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