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Mirrors > Home > MPE Home > Th. List > Mathboxes > usgr0e | Structured version Visualization version GIF version |
Description: The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
usgr0e.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
usgr0e.e | ⊢ (𝜑 → (iEdg‘𝐺) = ∅) |
Ref | Expression |
---|---|
usgr0e | ⊢ (𝜑 → 𝐺 ∈ USGraph ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr0e.e | . . 3 ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | |
2 | 1 | f10d 6082 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2}) |
3 | usgr0e.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
4 | eqid 2610 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
6 | 4, 5 | isusgr 40383 | . . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})) |
7 | 3, 6 | syl 17 | . 2 ⊢ (𝜑 → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})) |
8 | 2, 7 | mpbird 246 | 1 ⊢ (𝜑 → 𝐺 ∈ USGraph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 dom cdm 5038 –1-1→wf1 5801 ‘cfv 5804 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 USGraph cusgr 40379 |
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-sbc 3403 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-uni 4373 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fv 5812 df-usgr 40381 |
This theorem is referenced by: usgr0vb 40463 uhgr0vusgr 40468 usgr0eop 40472 edg0usgr 40479 usgr1v 40482 griedg0ssusgr 40489 cusgr1v 40653 frgr0v 41433 |
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