Theorem prcliscplgr 29449

Description: A proper class (representing a null graph, see vtxvalprc 29080) has the property of a complete graph (see also cplgr0v 29462), 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 29450. (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 6912	. 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2		cplgruvtxb.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	eqeq1i 2745	. . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4		rzal 4532	. . 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 2108  ∀wral 3067  Vcvv 3488  ∅c0 4352  ‘cfv 6573  Vtxcvtx 29031  UnivVtxcuvtx 29420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581

This theorem is referenced by: (None)