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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* ∅RegUSGraph𝑘 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5064 | . 2 ⊢ ∅ ∈ V | |
2 | vtxval0 26542 | . 2 ⊢ (Vtx‘∅) = ∅ | |
3 | iedgval0 26543 | . 2 ⊢ (iEdg‘∅) = ∅ | |
4 | 0vtxrusgr 27077 | . 2 ⊢ ((∅ ∈ V ∧ (Vtx‘∅) = ∅ ∧ (iEdg‘∅) = ∅) → ∀𝑘 ∈ ℕ0* ∅RegUSGraph𝑘) | |
5 | 1, 2, 3, 4 | mp3an 1441 | 1 ⊢ ∀𝑘 ∈ ℕ0* ∅RegUSGraph𝑘 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 ∈ wcel 2051 ∀wral 3081 Vcvv 3408 ∅c0 4172 class class class wbr 4925 ‘cfv 6185 ℕ0*cxnn0 11777 Vtxcvtx 26499 iEdgciedg 26500 RegUSGraphcrusgr 27056 |
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 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-i2m1 10401 ax-1ne0 10402 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 |
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 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-2 11501 df-slot 16341 df-base 16343 df-edgf 26493 df-vtx 26501 df-iedg 26502 df-uhgr 26561 df-upgr 26585 df-uspgr 26653 df-usgr 26654 df-rgr 27057 df-rusgr 27058 |
This theorem is referenced by: 0grrgr 27080 |
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