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Mirrors > Home > MPE Home > Th. List > 0grrgr | Structured version Visualization version GIF version |
Description: The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
0grrgr | ⊢ ∀𝑘 ∈ ℕ0* ∅RegGraph𝑘 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0grrusgr 27064 | . 2 ⊢ ∀𝑘 ∈ ℕ0* ∅RegUSGraph𝑘 | |
2 | rusgrrgr 27048 | . . 3 ⊢ (∅RegUSGraph𝑘 → ∅RegGraph𝑘) | |
3 | 2 | ralimi 3111 | . 2 ⊢ (∀𝑘 ∈ ℕ0* ∅RegUSGraph𝑘 → ∀𝑘 ∈ ℕ0* ∅RegGraph𝑘) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ∀𝑘 ∈ ℕ0* ∅RegGraph𝑘 |
Colors of variables: wff setvar class |
Syntax hints: ∀wral 3089 ∅c0 4179 class class class wbr 4929 ℕ0*cxnn0 11779 RegGraphcrgr 27040 RegUSGraphcrusgr 27041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-i2m1 10403 ax-1ne0 10404 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-2 11503 df-slot 16343 df-base 16345 df-edgf 26478 df-vtx 26486 df-iedg 26487 df-uhgr 26546 df-upgr 26570 df-uspgr 26638 df-usgr 26639 df-rgr 27042 df-rusgr 27043 |
This theorem is referenced by: (None) |
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