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Theorem prcliscplgr 27214
 Description: A proper class (representing a null graph, see vtxvalprc 26848) has the property of a complete graph (see also cplgr0v 27227), 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 27215. (Contributed by AV, 14-Feb-2022.)
Hypothesis
Ref Expression
cplgruvtxb.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
prcliscplgr 𝐺 ∈ V → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
Distinct variable groups:   𝑣,𝐺   𝑣,𝑉

Proof of Theorem prcliscplgr
StepHypRef Expression
1 fvprc 6639 . 2 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2 cplgruvtxb.v . . . 4 𝑉 = (Vtx‘𝐺)
32eqeq1i 2803 . . 3 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4 rzal 4411 . . 3 (𝑉 = ∅ → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
53, 4sylbir 238 . 2 ((Vtx‘𝐺) = ∅ → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
61, 5syl 17 1 𝐺 ∈ V → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ∅c0 4243  ‘cfv 6325  Vtxcvtx 26799  UnivVtxcuvtx 27185 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5175  ax-pow 5232 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333 This theorem is referenced by: (None)
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