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Theorem cplgr0v 27220
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 4436 . . 3 (𝑉 = ∅ → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
21adantl 485 . 2 ((𝐺𝑊𝑉 = ∅) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
3 cplgr0v.v . . . 4 𝑉 = (Vtx‘𝐺)
43iscplgr 27208 . . 3 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
54adantr 484 . 2 ((𝐺𝑊𝑉 = ∅) → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
62, 5mpbird 260 1 ((𝐺𝑊𝑉 = ∅) → 𝐺 ∈ ComplGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  c0 4276  cfv 6343  Vtxcvtx 26792  UnivVtxcuvtx 27178  ComplGraphccplgr 27202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-uvtx 27179  df-cplgr 27204
This theorem is referenced by:  cusgr0v  27221
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