![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > usgr1v0e | Structured version Visualization version GIF version |
Description: The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.) |
Ref | Expression |
---|---|
fusgredgfi.v | ⊢ 𝑉 = (Vtx‘𝐺) |
fusgredgfi.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
usgr1v0e | ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐺 ∈ USGraph) | |
2 | vex 3482 | . . . . . . . 8 ⊢ 𝑣 ∈ V | |
3 | fusgredgfi.v | . . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | eqeq1i 2740 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑣} ↔ (Vtx‘𝐺) = {𝑣}) |
5 | 4 | biimpi 216 | . . . . . . . . 9 ⊢ (𝑉 = {𝑣} → (Vtx‘𝐺) = {𝑣}) |
6 | 5 | adantl 481 | . . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (Vtx‘𝐺) = {𝑣}) |
7 | usgr1vr 29287 | . . . . . . . 8 ⊢ ((𝑣 ∈ V ∧ (Vtx‘𝐺) = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) | |
8 | 2, 6, 7 | sylancr 587 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) |
9 | 1, 8 | mpd 15 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (iEdg‘𝐺) = ∅) |
10 | fusgredgfi.e | . . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺) | |
11 | 10 | eqeq1i 2740 | . . . . . . 7 ⊢ (𝐸 = ∅ ↔ (Edg‘𝐺) = ∅) |
12 | usgruhgr 29218 | . . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UHGraph) | |
13 | uhgriedg0edg0 29159 | . . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | |
14 | 12, 13 | syl 17 | . . . . . . . 8 ⊢ (𝐺 ∈ USGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
15 | 14 | adantr 480 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
16 | 11, 15 | bitrid 283 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → (𝐸 = ∅ ↔ (iEdg‘𝐺) = ∅)) |
17 | 9, 16 | mpbird 257 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑉 = {𝑣}) → 𝐸 = ∅) |
18 | 17 | ex 412 | . . . 4 ⊢ (𝐺 ∈ USGraph → (𝑉 = {𝑣} → 𝐸 = ∅)) |
19 | 18 | exlimdv 1931 | . . 3 ⊢ (𝐺 ∈ USGraph → (∃𝑣 𝑉 = {𝑣} → 𝐸 = ∅)) |
20 | 3 | fvexi 6921 | . . . 4 ⊢ 𝑉 ∈ V |
21 | hash1snb 14455 | . . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) | |
22 | 20, 21 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣})) |
23 | 10 | fvexi 6921 | . . . 4 ⊢ 𝐸 ∈ V |
24 | hasheq0 14399 | . . . 4 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) | |
25 | 23, 24 | mp1i 13 | . . 3 ⊢ (𝐺 ∈ USGraph → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) |
26 | 19, 22, 25 | 3imtr4d 294 | . 2 ⊢ (𝐺 ∈ USGraph → ((♯‘𝑉) = 1 → (♯‘𝐸) = 0)) |
27 | 26 | imp 406 | 1 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 ‘cfv 6563 0cc0 11153 1c1 11154 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 Edgcedg 29079 UHGraphcuhgr 29088 USGraphcusgr 29181 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-edg 29080 df-uhgr 29090 df-upgr 29114 df-uspgr 29182 df-usgr 29183 |
This theorem is referenced by: cusgrsizeindb1 29483 |
Copyright terms: Public domain | W3C validator |