Theorem iscusgrvtx 40643

Description: A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.)

Hypothesis

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

Assertion

Ref	Expression
iscusgrvtx	⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑉

Proof of Theorem iscusgrvtx

Step	Hyp	Ref	Expression
1		iscusgr 40640	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2		iscusgrvtx.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	iscplgr 40636	. . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
4	3	pm5.32i 667	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
5	1, 4	bitri 263	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  Vtxcvtx 25673   USGraph cusgr 40379  UnivVtxcuvtxa 40551  ComplGraphccplgr 40552  ComplUSGraphccusgr 40553

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  df-cusgr 40558

This theorem is referenced by: (None)