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Mirrors > Home > MPE Home > Th. List > usgr0vb | Structured version Visualization version GIF version |
Description: The null graph, with no vertices, is a simple graph iff the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 16-Oct-2020.) |
Ref | Expression |
---|---|
usgr0vb | ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgruhgr 28947 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
2 | uhgr0vb 28836 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) | |
3 | 1, 2 | imbitrid 243 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
4 | simpl 482 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊) | |
5 | simpr 484 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅) | |
6 | 4, 5 | usgr0e 28997 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph) |
7 | 6 | ex 412 | . . 3 ⊢ (𝐺 ∈ 𝑊 → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ USGraph)) |
8 | 7 | adantr 480 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ USGraph)) |
9 | 3, 8 | impbid 211 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∅c0 4317 ‘cfv 6536 Vtxcvtx 28760 iEdgciedg 28761 UHGraphcuhgr 28820 USGraphcusgr 28913 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-i2m1 11177 ax-1ne0 11178 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-2 12276 df-uhgr 28822 df-upgr 28846 df-uspgr 28914 df-usgr 28915 |
This theorem is referenced by: usgr0v 29002 usgr1v 29017 |
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