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 ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑉

Proof of Theorem iscusgrvtx

Step	Hyp	Ref	Expression
1		iscusgr 29275	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2		iscusgrvtx.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	iscplgr 29272	. . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
4	3	pm5.32i 573	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
5	1, 4	bitri 274	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

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)