Theorem 0vconngr 27569

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 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph)

Proof of Theorem 0vconngr

Dummy variables 𝑓 𝑘 𝑛 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rzal 4295	. . 3 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)
2	1	adantl 475	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)
3		eqid 2825	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	isconngr 27565	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
5	4	adantr 474	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
6	2, 5	mpbird 249	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658  ∃wex 1880   ∈ wcel 2166  ∀wral 3117  ∅c0 4144   class class class wbr 4873  ‘cfv 6123  (class class class)co 6905  Vtxcvtx 26294  PathsOncpthson 27016  ConnGraphcconngr 27562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-conngr 27563

This theorem is referenced by:  1conngr  27570