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Theorem prcliscplgr 29140
Description: A proper class (representing a null graph, see vtxvalprc 28774) has the property of a complete graph (see also cplgr0v 29153), 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 29141. (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 6873 . 2 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
2 cplgruvtxb.v . . . 4 𝑉 = (Vtx‘𝐺)
32eqeq1i 2729 . . 3 (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅)
4 rzal 4500 . . 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 1533  wcel 2098  wral 3053  Vcvv 3466  c0 4314  cfv 6533  Vtxcvtx 28725  UnivVtxcuvtx 29111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541
This theorem is referenced by: (None)
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