Theorem cplgr0v 29518

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 4425	. . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
2	1	adantl 483	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
3		cplgr0v.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	iscplgr 29506	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
5	4	adantr 482	. 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 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∅c0 4264  ‘cfv 6489  Vtxcvtx 29087  UnivVtxcuvtx 29476  ComplGraphccplgr 29500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7363  df-uvtx 29477  df-cplgr 29502

This theorem is referenced by:  cusgr0v  29519