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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 27136) has the property of a complete graph (see also cplgr0v 27515), 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 27503. (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 6709 | . 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
2 | cplgruvtxb.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | eqeq1i 2742 | . . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
4 | rzal 4420 | . . 3 ⊢ (𝑉 = ∅ → ∀𝑣 ∈ 𝑉 𝑣 ∈ (UnivVtx‘𝐺)) | |
5 | 3, 4 | sylbir 238 | . 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 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ∅c0 4237 ‘cfv 6380 Vtxcvtx 27087 UnivVtxcuvtx 27473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 |
This theorem is referenced by: (None) |
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