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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 28245 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) | |
2 | 1 | adantrl 715 | . . . 4 ⊢ ((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
3 | simplrl 776 | . . . . . 6 ⊢ (((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ 𝑊) | |
4 | simpr 486 | . . . . . 6 ⊢ (((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) ∧ (iEdg‘𝐺) = ∅) → (iEdg‘𝐺) = ∅) | |
5 | 3, 4 | usgr0e 28226 | . . . . 5 ⊢ (((𝐴 ∈ V ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph) |
6 | 5 | ex 414 | . . . 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 414 | . 2 ⊢ (𝐴 ∈ V → ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅))) |
9 | snprc 4679 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
10 | simpl 484 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → 𝐺 ∈ 𝑊) | |
11 | simprr 772 | . . . . . 6 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (Vtx‘𝐺) = {𝐴}) | |
12 | simpl 484 | . . . . . 6 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → {𝐴} = ∅) | |
13 | 11, 12 | eqtrd 2773 | . . . . 5 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (Vtx‘𝐺) = ∅) |
14 | usgr0vb 28227 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) | |
15 | 10, 13, 14 | syl2an2 685 | . . . 4 ⊢ (({𝐴} = ∅ ∧ (𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴})) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) |
16 | 15 | ex 414 | . . 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 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 {csn 4587 ‘cfv 6497 Vtxcvtx 27989 iEdgciedg 27990 USGraphcusgr 28142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-dju 9842 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-xnn0 12491 df-z 12505 df-uz 12769 df-fz 13431 df-hash 14237 df-edg 28041 df-uhgr 28051 df-upgr 28075 df-uspgr 28143 df-usgr 28144 |
This theorem is referenced by: usgr1v0edg 28247 |
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