Theorem iscusgrvtx 29405

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 29402	. 2 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
2		iscusgrvtx.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	iscplgr 29399	. . 3 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
4	3	pm5.32i 574	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph) ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
5	1, 4	bitri 275	1 ⊢ (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ‘cfv 6536  Vtxcvtx 28980  USGraphcusgr 29133  UnivVtxcuvtx 29369  ComplGraphccplgr 29393  ComplUSGraphccusgr 29394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-uvtx 29370  df-cplgr 29395  df-cusgr 29396

This theorem is referenced by: (None)