Theorem 0conngra 26202

Description: A class/graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)

Assertion

Ref	Expression
0conngra	⊢ (𝐸 ∈ 𝑉 → ∅ ConnGrph 𝐸)

Proof of Theorem 0conngra

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

Step	Hyp	Ref	Expression
1		0ex 4718	. 2 ⊢ ∅ ∈ V
2		ral0 4028	. . 3 ⊢ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓∃𝑝 𝑓(𝑘(∅ PathOn 𝐸)𝑛)𝑝
3		isconngra 26200	. . 3 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ 𝑉) → (∅ ConnGrph 𝐸 ↔ ∀𝑘 ∈ ∅ ∀𝑛 ∈ ∅ ∃𝑓∃𝑝 𝑓(𝑘(∅ PathOn 𝐸)𝑛)𝑝))
4	2, 3	mpbiri 247	. 2 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ 𝑉) → ∅ ConnGrph 𝐸)
5	1, 4	mpan 702	1 ⊢ (𝐸 ∈ 𝑉 → ∅ ConnGrph 𝐸)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874   class class class wbr 4583  (class class class)co 6549   PathOn cpthon 26032   ConnGrph cconngra 26197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-conngra 26198

This theorem is referenced by:  1conngra  26203