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Theorem cusgruvtxb 28946
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 28942 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2 ibar 527 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph)))
3 iscusgrvtx.v . . . 4 𝑉 = (Vtx‘𝐺)
43cplgruvtxb 28937 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
52, 4bitr3d 280 . 2 (𝐺 ∈ USGraph → ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (UnivVtx‘𝐺) = 𝑉))
61, 5bitrid 282 1 (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  cfv 6542  Vtxcvtx 28523  USGraphcusgr 28676  UnivVtxcuvtx 28909  ComplGraphccplgr 28933  ComplUSGraphccusgr 28934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-cplgr 28935  df-cusgr 28936
This theorem is referenced by:  vdiscusgrb  29054  vdiscusgr  29055
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