Theorem prcliscplgr 29431

Description: A proper class (representing a null graph, see vtxvalprc 29062) has the property of a complete graph (see also cplgr0v 29444), 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 29432. (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 6898	. 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	eqeq1i 2742	. . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4		rzal 4509	. . 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 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  ∅c0 4333  ‘cfv 6561  Vtxcvtx 29013  UnivVtxcuvtx 29402

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569

This theorem is referenced by: (None)