Theorem prcliscplgr 29446

Description: A proper class (representing a null graph, see vtxvalprc 29077) has the property of a complete graph (see also cplgr0v 29459), 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 29447. (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 6899	. 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	eqeq1i 2740	. . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4		rzal 4515	. . 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 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  ∅c0 4339  ‘cfv 6563  Vtxcvtx 29028  UnivVtxcuvtx 29417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571

This theorem is referenced by: (None)