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Mirrors > Home > MPE Home > Th. List > 1hevtxdg0 | Structured version Visualization version GIF version |
Description: The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 is 0 if 𝐷 is not incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) |
Ref | Expression |
---|---|
1hevtxdg0.i | ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) |
1hevtxdg0.v | ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) |
1hevtxdg0.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
1hevtxdg0.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
1hevtxdg0.e | ⊢ (𝜑 → 𝐸 ∈ 𝑌) |
1hevtxdg0.n | ⊢ (𝜑 → 𝐷 ∉ 𝐸) |
Ref | Expression |
---|---|
1hevtxdg0 | ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1hevtxdg0.n | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∉ 𝐸) | |
2 | df-nel 2898 | . . . . . . 7 ⊢ (𝐷 ∉ 𝐸 ↔ ¬ 𝐷 ∈ 𝐸) | |
3 | 1, 2 | sylib 208 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐷 ∈ 𝐸) |
4 | 1hevtxdg0.i | . . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) | |
5 | 4 | fveq1d 6193 | . . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐴) = ({〈𝐴, 𝐸〉}‘𝐴)) |
6 | 1hevtxdg0.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
7 | 1hevtxdg0.e | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑌) | |
8 | fvsng 6447 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ({〈𝐴, 𝐸〉}‘𝐴) = 𝐸) | |
9 | 6, 7, 8 | syl2anc 693 | . . . . . . 7 ⊢ (𝜑 → ({〈𝐴, 𝐸〉}‘𝐴) = 𝐸) |
10 | 5, 9 | eqtrd 2656 | . . . . . 6 ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐴) = 𝐸) |
11 | 3, 10 | neleqtrrd 2723 | . . . . 5 ⊢ (𝜑 → ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴)) |
12 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘𝐴)) | |
13 | 12 | eleq2d 2687 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
14 | 13 | notbid 308 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
15 | 14 | ralsng 4218 | . . . . . 6 ⊢ (𝐴 ∈ 𝑋 → (∀𝑥 ∈ {𝐴} ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
16 | 6, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ {𝐴} ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝐴))) |
17 | 11, 16 | mpbird 247 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) |
18 | 4 | dmeqd 5326 | . . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom {〈𝐴, 𝐸〉}) |
19 | dmsnopg 5606 | . . . . . . 7 ⊢ (𝐸 ∈ 𝑌 → dom {〈𝐴, 𝐸〉} = {𝐴}) | |
20 | 7, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom {〈𝐴, 𝐸〉} = {𝐴}) |
21 | 18, 20 | eqtrd 2656 | . . . . 5 ⊢ (𝜑 → dom (iEdg‘𝐺) = {𝐴}) |
22 | 21 | raleqdv 3144 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ∀𝑥 ∈ {𝐴} ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥))) |
23 | 17, 22 | mpbird 247 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) |
24 | ralnex 2992 | . . 3 ⊢ (∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝐷 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) | |
25 | 23, 24 | sylib 208 | . 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥)) |
26 | 1hevtxdg0.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
27 | 1hevtxdg0.v | . . . . 5 ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | |
28 | 27 | eleq2d 2687 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ (Vtx‘𝐺) ↔ 𝐷 ∈ 𝑉)) |
29 | 26, 28 | mpbird 247 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (Vtx‘𝐺)) |
30 | eqid 2622 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
31 | eqid 2622 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
32 | eqid 2622 | . . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
33 | 30, 31, 32 | vtxd0nedgb 26384 | . . 3 ⊢ (𝐷 ∈ (Vtx‘𝐺) → (((VtxDeg‘𝐺)‘𝐷) = 0 ↔ ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥))) |
34 | 29, 33 | syl 17 | . 2 ⊢ (𝜑 → (((VtxDeg‘𝐺)‘𝐷) = 0 ↔ ¬ ∃𝑥 ∈ dom (iEdg‘𝐺)𝐷 ∈ ((iEdg‘𝐺)‘𝑥))) |
35 | 25, 34 | mpbird 247 | 1 ⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 ∀wral 2912 ∃wrex 2913 {csn 4177 〈cop 4183 dom cdm 5114 ‘cfv 5888 0cc0 9936 Vtxcvtx 25874 iEdgciedg 25875 VtxDegcvtxdg 26361 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
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-reu 2919 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-xadd 11947 df-fz 12327 df-hash 13118 df-vtxdg 26362 |
This theorem is referenced by: p1evtxdeq 26409 eupth2lem3lem6 27093 |
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