Theorem prcliscplgr 29390

Description: A proper class (representing a null graph, see vtxvalprc 29021) has the property of a complete graph (see also cplgr0v 29403), but cannot be an element of ComplGraph, of course. Because of this, a sethood antecedent like 𝐺 ∈ 𝑊 is necessary in the following theorems like iscplgr 29391. (Contributed by AV, 14-Feb-2022.)

Hypothesis

Ref	Expression
cplgruvtxb.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
prcliscplgr	⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑉

Proof of Theorem prcliscplgr

Step	Hyp	Ref	Expression
1		fvprc 6814	. 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	eqeq1i 2736	. . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4		rzal 4459	. . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
5	3, 4	sylbir 235	. 2 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))
6	1, 5	syl 17	1 ⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436  ∅c0 4283  ‘cfv 6481  Vtxcvtx 28972  UnivVtxcuvtx 29361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489

This theorem is referenced by: (None)