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Mirrors > Home > MPE Home > Th. List > usgr1v | Structured version Visualization version GIF version |
Description: A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.) |
Ref | Expression |
---|---|
usgr1v | ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr1vr 29006 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) | |
2 | 1 | adantrl 713 | . . . 4 ⊢ ((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
3 | simplrl 774 | . . . . . 6 ⊢ (((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊) | |
4 | simpr 484 | . . . . . 6 ⊢ (((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅) | |
5 | 3, 4 | usgr0e 28987 | . . . . 5 ⊢ (((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph) |
6 | 5 | ex 412 | . . . 4 ⊢ ((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → ((iEdg‘𝐺) = ∅ → 𝐺 ∈ USGraph)) |
7 | 2, 6 | impbid 211 | . . 3 ⊢ ((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) |
8 | 7 | ex 412 | . 2 ⊢ (𝐴 ∈ V → ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))) |
9 | snprc 4714 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | simpl 482 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → 𝐺 ∈ 𝑊) | |
11 | simprr 770 | . . . . . 6 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (Vtx‘𝐺) = {𝐴}) | |
12 | simpl 482 | . . . . . 6 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → {𝐴} = ∅) | |
13 | 11, 12 | eqtrd 2764 | . . . . 5 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (Vtx‘𝐺) = ∅) |
14 | usgr0vb 28988 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) | |
15 | 10, 13, 14 | syl2an2 683 | . . . 4 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) |
16 | 15 | ex 412 | . . 3 ⊢ ({𝐴} = ∅ → ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))) |
17 | 9, 16 | sylbi 216 | . 2 ⊢ (¬ 𝐴 ∈ V → ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))) |
18 | 8, 17 | pm2.61i 182 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∅c0 4315 {csn 4621 ‘cfv 6534 Vtxcvtx 28750 iEdgciedg 28751 USGraphcusgr 28903 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-dju 9893 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-fz 13486 df-hash 14292 df-edg 28802 df-uhgr 28812 df-upgr 28836 df-uspgr 28904 df-usgr 28905 |
This theorem is referenced by: usgr1v0edg 29008 |
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