0grrgr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 0grrgr		Structured version Visualization version GIF version

Theorem 0grrgr 26476

Description: The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.)

**Assertion**
Ref	Expression
0grrgr	⊢ ∀𝑘 ∈ ℕ₀^* ∅ RegGraph 𝑘

**Proof of Theorem 0grrgr**
Step	Hyp	Ref	Expression
1		0grrusgr 26475	. 2 ⊢ ∀𝑘 ∈ ℕ₀^* ∅ RegUSGraph 𝑘
2		rusgrrgr 26459	. . 3 ⊢ (∅ RegUSGraph 𝑘 → ∅ RegGraph 𝑘)
3	2	ralimi 2952	. 2 ⊢ (∀𝑘 ∈ ℕ₀^* ∅ RegUSGraph 𝑘 → ∀𝑘 ∈ ℕ₀^* ∅ RegGraph 𝑘)
4	1, 3	ax-mp 5	1 ⊢ ∀𝑘 ∈ ℕ₀^* ∅ RegGraph 𝑘

Colors of variables: wff setvar class

Syntax hints: ∀wral 2912 ∅c0 3915 class class class wbr 4653 ℕ₀^*cxnn0 11363 RegGraph crgr 26451 RegUSGraph crusgr 26452

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-2 11079 df-slot 15861 df-base 15863 df-edgf 25868 df-vtx 25876 df-iedg 25877 df-uhgr 25953 df-upgr 25977 df-uspgr 26045 df-usgr 26046 df-rgr 26453 df-rusgr 26454

This theorem is referenced by: (None)