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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 483 | . . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝑣 ∈ (Vtx‘𝐺)) | |
2 | simplr 767 | . . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (iEdg‘𝐺) = ∅) | |
3 | eqid 2725 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | eqid 2725 | . . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 3, 4 | vtxdg0e 29360 | . . . . 5 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (iEdg‘𝐺) = ∅) → ((VtxDeg‘𝐺)‘𝑣) = 0) |
6 | 1, 2, 5 | syl2anc 582 | . . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) ∧ 𝑣 ∈ (Vtx‘𝐺)) → ((VtxDeg‘𝐺)‘𝑣) = 0) |
7 | 6 | ralrimiva 3135 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 0) |
8 | 0xnn0 12583 | . . 3 ⊢ 0 ∈ ℕ0* | |
9 | 7, 8 | jctil 518 | . 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 2725 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
12 | 3, 11 | isrgr 29445 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 0 ∈ ℕ0*) → (𝐺 RegGraph 0 ↔ (0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 0))) |
13 | 10, 12 | sylan2 591 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → (𝐺 RegGraph 0 ↔ (0 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)((VtxDeg‘𝐺)‘𝑣) = 0))) |
14 | 9, 13 | mpbird 256 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 RegGraph 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∅c0 4322 class class class wbr 5149 ‘cfv 6549 0cc0 11140 ℕ0*cxnn0 12577 Vtxcvtx 28881 iEdgciedg 28882 VtxDegcvtxdg 29351 RegGraph crgr 29441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-xadd 13128 df-fz 13520 df-hash 14326 df-vtxdg 29352 df-rgr 29443 |
This theorem is referenced by: uhgr0edg0rgr 29459 |
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