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Theorem prcliscplgr 27123
Description: A proper class (representing a null graph, see vtxvalprc 26757) has the property of a complete graph (see also cplgr0v 27136), 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 27124. (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 6656 . 2 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2 cplgruvtxb.v . . . 4 𝑉 = (Vtx‘𝐺)
32eqeq1i 2823 . . 3 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4 rzal 4449 . . 3 (𝑉 = ∅ → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
53, 4sylbir 236 . 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 1528  wcel 2105  wral 3135  Vcvv 3492  c0 4288  cfv 6348  Vtxcvtx 26708  UnivVtxcuvtx 27094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201  ax-pow 5257
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by: (None)
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