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Theorem prcliscplgr 28651
Description: A proper class (representing a null graph, see vtxvalprc 28285) has the property of a complete graph (see also cplgr0v 28664), but cannot be an element of ComplGraph, of course. Because of this, a sethood antecedent like 𝐺𝑊 is necessary in the following theorems like iscplgr 28652. (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 6880 . 2 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2 cplgruvtxb.v . . . 4 𝑉 = (Vtx‘𝐺)
32eqeq1i 2738 . . 3 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4 rzal 4507 . . 3 (𝑉 = ∅ → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
53, 4sylbir 234 . 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 1542  wcel 2107  wral 3062  Vcvv 3475  c0 4321  cfv 6540  Vtxcvtx 28236  UnivVtxcuvtx 28622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548
This theorem is referenced by: (None)
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