Theorem cplgr0v 29391

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 4462	. . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
2	1	adantl 481	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
3		cplgr0v.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	iscplgr 29379	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
5	4	adantr 480	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
6	2, 5	mpbird 257	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ ComplGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∅c0 4286  ‘cfv 6486  Vtxcvtx 28960  UnivVtxcuvtx 29349  ComplGraphccplgr 29373

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 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-uvtx 29350  df-cplgr 29375

This theorem is referenced by:  cusgr0v  29392