Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cplgr0v Structured version   Visualization version   GIF version

Theorem cplgr0v 26792
 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 4296 . . 3 (𝑉 = ∅ → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
21adantl 475 . 2 ((𝐺𝑊𝑉 = ∅) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
3 cplgr0v.v . . . 4 𝑉 = (Vtx‘𝐺)
43iscplgr 26780 . . 3 (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
54adantr 474 . 2 ((𝐺𝑊𝑉 = ∅) → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
62, 5mpbird 249 1 ((𝐺𝑊𝑉 = ∅) → 𝐺 ∈ ComplGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∅c0 4141  ‘cfv 6137  Vtxcvtx 26361  UnivVtxcuvtx 26750  ComplGraphccplgr 26774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-uvtx 26751  df-cplgr 26776 This theorem is referenced by:  cusgr0v  26793
 Copyright terms: Public domain W3C validator