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Mirrors > Home > MPE Home > Th. List > uhgr0vsize0 | Structured version Visualization version GIF version |
Description: The size of a hypergraph with no vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 7-Nov-2020.) |
Ref | Expression |
---|---|
uhgr0v0e.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgr0v0e.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
uhgr0vsize0 | ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgr0v0e.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uhgr0v0e.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | uhgr0v0e 26739 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅) |
4 | 3 | ex 405 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → 𝐸 = ∅)) |
5 | 1 | fvexi 6511 | . . . 4 ⊢ 𝑉 ∈ V |
6 | hasheq0 13538 | . . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ((♯‘𝑉) = 0 ↔ 𝑉 = ∅) |
8 | 2 | fvexi 6511 | . . . 4 ⊢ 𝐸 ∈ V |
9 | hasheq0 13538 | . . . 4 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅) |
11 | 4, 7, 10 | 3imtr4g 288 | . 2 ⊢ (𝐺 ∈ UHGraph → ((♯‘𝑉) = 0 → (♯‘𝐸) = 0)) |
12 | 11 | imp 398 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 Vcvv 3410 ∅c0 4173 ‘cfv 6186 0cc0 10334 ♯chash 13504 Vtxcvtx 26500 Edgcedg 26551 UHGraphcuhgr 26560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-hash 13505 df-edg 26552 df-uhgr 26562 |
This theorem is referenced by: uhgr0edgfi 26741 cusgrsizeindb0 26950 |
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