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Mirrors > Home > MPE Home > Th. List > 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 6651 | . 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
3 | usgr0e.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
4 | eqid 2824 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | eqid 2824 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
6 | 4, 5 | isusgr 26941 | . . 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 259 | 1 ⊢ (𝜑 → 𝐺 ∈ USGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 {crab 3145 ∖ cdif 3936 ∅c0 4294 𝒫 cpw 4542 {csn 4570 dom cdm 5558 –1-1→wf1 6355 ‘cfv 6358 2c2 11695 ♯chash 13693 Vtxcvtx 26784 iEdgciedg 26785 USGraphcusgr 26937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fv 6366 df-usgr 26939 |
This theorem is referenced by: usgr0vb 27022 uhgr0vusgr 27027 usgr0eop 27031 edg0usgr 27038 usgr1v 27041 griedg0ssusgr 27050 cusgr1v 27216 frgr0v 28044 |
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