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

Theorem iscusgrvtx 29278
Description: A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.)
Hypothesis
Ref Expression
iscusgrvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
iscusgrvtx (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝑉

Proof of Theorem iscusgrvtx
StepHypRef Expression
1 iscusgr 29275 . 2 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2 iscusgrvtx.v . . . 4 𝑉 = (Vtx‘𝐺)
32iscplgr 29272 . . 3 (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
43pm5.32i 573 . 2 ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
51, 4bitri 274 1 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3051  cfv 6543  Vtxcvtx 28853  USGraphcusgr 29006  UnivVtxcuvtx 29242  ComplGraphccplgr 29266  ComplUSGraphccusgr 29267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-uvtx 29243  df-cplgr 29268  df-cusgr 29269
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator