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Mirrors > Home > MPE Home > Th. List > usgr1vr | Structured version Visualization version GIF version |
Description: A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 2-Apr-2021.) |
Ref | Expression |
---|---|
usgr1vr | ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgruhgr 26962 | . . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
2 | 1 | adantl 484 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ UHGraph) |
3 | fveq2 6664 | . . . . . 6 ⊢ ((Vtx‘𝐺) = {𝐴} → (♯‘(Vtx‘𝐺)) = (♯‘{𝐴})) | |
4 | hashsng 13724 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (♯‘{𝐴}) = 1) | |
5 | 3, 4 | sylan9eqr 2878 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (♯‘(Vtx‘𝐺)) = 1) |
6 | 5 | adantr 483 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (♯‘(Vtx‘𝐺)) = 1) |
7 | eqid 2821 | . . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | eqid 2821 | . . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
9 | 7, 8 | usgrislfuspgr 26963 | . . . . . 6 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)})) |
10 | 9 | simprbi 499 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
11 | 10 | adantl 484 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
12 | eqid 2821 | . . . . 5 ⊢ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)} = {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)} | |
13 | 7, 8, 12 | lfuhgr1v0e 27030 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘(Vtx‘𝐺)) = 1 ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) → (Edg‘𝐺) = ∅) |
14 | 2, 6, 11, 13 | syl3anc 1367 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (Edg‘𝐺) = ∅) |
15 | uhgriedg0edg0 26906 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
16 | 1, 15 | syl 17 | . . . 4 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
17 | 16 | adantl 484 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
18 | 14, 17 | mpbid 234 | . 2 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (iEdg‘𝐺) = ∅) |
19 | 18 | ex 415 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ∅c0 4290 𝒫 cpw 4538 {csn 4560 class class class wbr 5058 dom cdm 5549 ⟶wf 6345 ‘cfv 6349 1c1 10532 ≤ cle 10670 2c2 11686 ♯chash 13684 Vtxcvtx 26775 iEdgciedg 26776 Edgcedg 26826 UHGraphcuhgr 26835 USPGraphcuspgr 26927 USGraphcusgr 26928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 df-edg 26827 df-uhgr 26837 df-upgr 26861 df-uspgr 26929 df-usgr 26930 |
This theorem is referenced by: usgr1v 27032 usgr1v0e 27102 |
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