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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 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 𝐺 ∈ 𝑊 is necessary in the following theorems like iscplgr 29141. (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 6873 | . 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅) | |
2 | cplgruvtxb.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | eqeq1i 2729 | . . 3 ⊢ (𝑉 = ∅ ↔ (Vtx‘𝐺) = ∅) |
4 | rzal 4500 | . . 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 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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