Theorem 0vconngr 30225

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 4532	. . 3 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)
2	1	adantl 481	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)
3		eqid 2740	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	isconngr 30221	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑛 ∈ (Vtx‘𝐺)∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
5	4	adantr 480	. 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 257	1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∅c0 4352   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Vtxcvtx 29031  PathsOncpthson 29750  ConnGraphcconngr 30218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-conngr 30219

This theorem is referenced by:  1conngr  30226