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Mirrors > Home > MPE Home > Th. List > uvtx0 | Structured version Visualization version GIF version |
Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.) |
Ref | Expression |
---|---|
uvtxel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
uvtx0 | ⊢ (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvtxel.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | uvtxval 29193 | . 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} |
3 | rabeq 3442 | . . 3 ⊢ (𝑉 = ∅ → {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}) | |
4 | rab0 4378 | . . 3 ⊢ {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅ | |
5 | 3, 4 | eqtrdi 2784 | . 2 ⊢ (𝑉 = ∅ → {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅) |
6 | 2, 5 | eqtrid 2780 | 1 ⊢ (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3057 {crab 3428 ∖ cdif 3942 ∅c0 4318 {csn 4624 ‘cfv 6542 (class class class)co 7414 Vtxcvtx 28802 NeighbVtx cnbgr 29138 UnivVtxcuvtx 29191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-uvtx 29192 |
This theorem is referenced by: (None) |
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