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Mirrors > Home > MPE Home > Th. List > usgra0 | Structured version Visualization version GIF version |
Description: The empty graph, with vertices but no edges, is a graph, analogous to umgra0 25854. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
usgra0 | ⊢ (𝑉 ∈ 𝑊 → 𝑉 USGrph ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2611 | . . 3 ⊢ (𝑉 ∈ 𝑊 → ∅ = ∅) | |
2 | 1 | f10d 6082 | . 2 ⊢ (𝑉 ∈ 𝑊 → ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) |
3 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
4 | isusgra 25873 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∅ ∈ V) → (𝑉 USGrph ∅ ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) | |
5 | 3, 4 | mpan2 703 | . 2 ⊢ (𝑉 ∈ 𝑊 → (𝑉 USGrph ∅ ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
6 | 2, 5 | mpbird 246 | 1 ⊢ (𝑉 ∈ 𝑊 → 𝑉 USGrph ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 –1-1→wf1 5801 ‘cfv 5804 2c2 10947 #chash 12979 USGrph cusg 25859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-usgra 25862 |
This theorem is referenced by: usgra0v 25900 usgra1v 25919 cusgra0v 25989 cusgra1v 25990 |
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