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Theorem cplgr0v 29459
Description: A null graph (with no vertices) 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 ((𝐺𝑊𝑉 = ∅) → 𝐺 ∈ ComplGraph)

Proof of Theorem cplgr0v
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 rzal 4515 . . 3 (𝑉 = ∅ → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
21adantl 481 . 2 ((𝐺𝑊𝑉 = ∅) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
3 cplgr0v.v . . . 4 𝑉 = (Vtx‘𝐺)
43iscplgr 29447 . . 3 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
54adantr 480 . 2 ((𝐺𝑊𝑉 = ∅) → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
62, 5mpbird 257 1 ((𝐺𝑊𝑉 = ∅) → 𝐺 ∈ ComplGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  c0 4339  cfv 6563  Vtxcvtx 29028  UnivVtxcuvtx 29417  ComplGraphccplgr 29441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-uvtx 29418  df-cplgr 29443
This theorem is referenced by:  cusgr0v  29460
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