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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 28015 | . . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅) |
4 | 3 | ex 413 | . . 3 ⊢ (𝐺 ∈ UHGraph → (𝑉 = ∅ → 𝐸 = ∅)) |
5 | 1 | fvexi 6853 | . . . 4 ⊢ 𝑉 ∈ V |
6 | hasheq0 14217 | . . . 4 ⊢ (𝑉 ∈ V → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ ((♯‘𝑉) = 0 ↔ 𝑉 = ∅) |
8 | 2 | fvexi 6853 | . . . 4 ⊢ 𝐸 ∈ V |
9 | hasheq0 14217 | . . . 4 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅) |
11 | 4, 7, 10 | 3imtr4g 295 | . 2 ⊢ (𝐺 ∈ UHGraph → ((♯‘𝑉) = 0 → (♯‘𝐸) = 0)) |
12 | 11 | imp 407 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 ‘cfv 6493 0cc0 11009 ♯chash 14184 Vtxcvtx 27776 Edgcedg 27827 UHGraphcuhgr 27836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-hash 14185 df-edg 27828 df-uhgr 27838 |
This theorem is referenced by: uhgr0edgfi 28017 cusgrsizeindb0 28226 |
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