Theorem cplgr0v 29630

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	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ ComplGraph)

Proof of Theorem cplgr0v

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rzal 4450	. . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
2	1	adantl 485	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
3		cplgr0v.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	iscplgr 29618	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
5	4	adantr 484	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
6	2, 5	mpbird 259	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ ComplGraph)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∅c0 4287  ‘cfv 6523  Vtxcvtx 29199  UnivVtxcuvtx 29588  ComplGraphccplgr 29612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-uvtx 29589  df-cplgr 29614

This theorem is referenced by:  cusgr0v  29631