Theorem cplgr0v 40649

Description: A graph with no vertices (and therefore no edges) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)

Hypothesis

Ref	Expression
cplgr0v.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
cplgr0v	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ ComplGraph)

Proof of Theorem cplgr0v

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4028	. . . 4 ⊢ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘𝐺)
2		raleq 3115	. . . 4 ⊢ (𝑉 = ∅ → (∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺) ↔ ∀𝑣 ∈ ∅ 𝑣 ∈ (UnivVtx‘𝐺)))
3	1, 2	mpbiri 247	. . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
4	3	adantl 481	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
5		cplgr0v.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
6	5	iscplgr 40636	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
7	6	adantr 480	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
8	4, 7	mpbird 246	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ ComplGraph)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874  ‘cfv 5804  Vtxcvtx 25673  UnivVtxcuvtxa 40551  ComplGraphccplgr 40552

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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

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-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-iota 5768  df-fv 5812  df-cplgr 40557

This theorem is referenced by:  cusgr0v  40650