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Theorem cusgruvtxb 40644
 Description: A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.)
Hypothesis
Ref Expression
iscusgrvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
cusgruvtxb (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Proof of Theorem cusgruvtxb
StepHypRef Expression
1 iscusgr 40640 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2 ibar 524 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)))
3 iscusgrvtx.v . . . 4 𝑉 = (Vtx‘𝐺)
43cplgruvtxb 40637 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
52, 4bitr3d 269 . 2 (𝐺 ∈ USGraph → ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (UnivVtx‘𝐺) = 𝑉))
61, 5syl5bb 271 1 (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘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-8 1979  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-pow 4769  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-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-uvtxa 40556  df-cplgr 40557  df-cusgr 40558 This theorem is referenced by:  vdiscusgrb  40746  vdiscusgr  40747
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