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Mirrors > Home > MPE Home > Th. List > 0grrusgr | Structured version Visualization version GIF version |
Description: The null graph represented by an empty set is a k-regular simple graph for every k. (Contributed by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
0grrusgr | ⊢ ∀𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5291 | . 2 ⊢ ∅ ∈ V | |
2 | vtxval0 28094 | . 2 ⊢ (Vtx‘∅) = ∅ | |
3 | iedgval0 28095 | . 2 ⊢ (iEdg‘∅) = ∅ | |
4 | 0vtxrusgr 28629 | . 2 ⊢ ((∅ ∈ V ∧ (Vtx‘∅) = ∅ ∧ (iEdg‘∅) = ∅) → ∀𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘) | |
5 | 1, 2, 3, 4 | mp3an 1461 | 1 ⊢ ∀𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3466 ∅c0 4309 class class class wbr 5132 ‘cfv 6523 ℕ0*cxnn0 12516 Vtxcvtx 28051 iEdgciedg 28052 RegUSGraph crusgr 28608 |
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 2702 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-ov 7387 df-om 7830 df-2nd 7949 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-er 8677 df-en 8913 df-dom 8914 df-sdom 8915 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-nn 12185 df-2 12247 df-3 12248 df-4 12249 df-5 12250 df-6 12251 df-7 12252 df-8 12253 df-9 12254 df-n0 12445 df-dec 12650 df-slot 17087 df-ndx 17099 df-base 17117 df-edgf 28042 df-vtx 28053 df-iedg 28054 df-uhgr 28113 df-upgr 28137 df-uspgr 28205 df-usgr 28206 df-rgr 28609 df-rusgr 28610 |
This theorem is referenced by: 0grrgr 28632 |
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