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Mirrors > Home > MPE Home > Th. List > prcliscplgr | Structured version Visualization version GIF version |
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 𝐺 ∈ 𝑊 is necessary in the following theorems like iscplgr 28652. (Contributed by AV, 14-Feb-2022.) |
Ref | Expression |
---|---|
cplgruvtxb.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
prcliscplgr | ⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvprc 6880 | . 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
2 | cplgruvtxb.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | eqeq1i 2738 | . . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
4 | rzal 4507 | . . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) | |
5 | 3, 4 | sylbir 234 | . 2 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) |
6 | 1, 5 | syl 17 | 1 ⊢ (¬ 𝐺 ∈ V → ∀𝑣 ∈ 𝑉 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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