Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 27551 | . . . . 5 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
2 | 1 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → 𝐺 ∈ UHGraph) |
3 | fveq2 6771 | . . . . . 6 ⊢ ((Vtx‘𝐺) = {𝐴} → (♯‘(Vtx‘𝐺)) = (♯‘{𝐴})) | |
4 | hashsng 14082 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (♯‘{𝐴}) = 1) | |
5 | 3, 4 | sylan9eqr 2802 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (♯‘(Vtx‘𝐺)) = 1) |
6 | 5 | adantr 481 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (♯‘(Vtx‘𝐺)) = 1) |
7 | eqid 2740 | . . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
8 | eqid 2740 | . . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
9 | 7, 8 | usgrislfuspgr 27552 | . . . . . 6 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)})) |
10 | 9 | simprbi 497 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
11 | 10 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)}) |
12 | eqid 2740 | . . . . 5 ⊢ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)} = {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 2 ≤ (♯‘𝑥)} | |
13 | 7, 8, 12 | lfuhgr1v0e 27619 | . . . 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 1370 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (Edg‘𝐺) = ∅) |
15 | uhgriedg0edg0 27495 | . . . . 5 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
16 | 1, 15 | syl 17 | . . . 4 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
17 | 16 | adantl 482 | . . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
18 | 14, 17 | mpbid 231 | . 2 ⊢ (((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) ∧ 𝐺 ∈ USGraph) → (iEdg‘𝐺) = ∅) |
19 | 18 | ex 413 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {crab 3070 ∅c0 4262 𝒫 cpw 4539 {csn 4567 class class class wbr 5079 dom cdm 5590 ⟶wf 6428 ‘cfv 6432 1c1 10873 ≤ cle 11011 2c2 12028 ♯chash 14042 Vtxcvtx 27364 iEdgciedg 27365 Edgcedg 27415 UHGraphcuhgr 27424 USPGraphcuspgr 27516 USGraphcusgr 27517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-oadd 8292 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-dju 9660 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12582 df-fz 13239 df-hash 14043 df-edg 27416 df-uhgr 27426 df-upgr 27450 df-uspgr 27518 df-usgr 27519 |
This theorem is referenced by: usgr1v 27621 usgr1v0e 27691 |
Copyright terms: Public domain | W3C validator |