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Mirrors > Home > MPE Home > Th. List > 0edg0rgr | Structured version Visualization version GIF version |
Description: A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
0edg0rgr | ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 RegGraph 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝑣 ∈ (Vtx‘𝐺)) | |
2 | simplr 769 | . . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (iEdg‘𝐺) = ∅) | |
3 | eqid 2736 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | eqid 2736 | . . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 3, 4 | vtxdg0e 27516 | . . . . 5 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (iEdg‘𝐺) = ∅) → ((VtxDeg‘𝐺)‘𝑣) = 0) |
6 | 1, 2, 5 | syl2anc 587 | . . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) ∧ 𝑣 ∈ (Vtx‘𝐺)) → ((VtxDeg‘𝐺)‘𝑣) = 0) |
7 | 6 | ralrimiva 3095 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 0) |
8 | 0xnn0 12133 | . . 3 ⊢ 0 ∈ ℕ0* | |
9 | 7, 8 | jctil 523 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 0)) |
10 | 8 | a1i 11 | . . 3 ⊢ ((iEdg‘𝐺) = ∅ → 0 ∈ ℕ0*) |
11 | eqid 2736 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
12 | 3, 11 | isrgr 27601 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 0 ∈ ℕ0*) → (𝐺 RegGraph 0 ↔ (0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 0))) |
13 | 10, 12 | sylan2 596 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (𝐺 RegGraph 0 ↔ (0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 0))) |
14 | 9, 13 | mpbird 260 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 RegGraph 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∅c0 4223 class class class wbr 5039 ‘cfv 6358 0cc0 10694 ℕ0*cxnn0 12127 Vtxcvtx 27041 iEdgciedg 27042 VtxDegcvtxdg 27507 RegGraph crgr 27597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-xadd 12670 df-fz 13061 df-hash 13862 df-vtxdg 27508 df-rgr 27599 |
This theorem is referenced by: uhgr0edg0rgr 27615 |
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