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Mirrors > Home > MPE Home > Th. List > vtxdusgr0edgnelALT | Structured version Visualization version GIF version |
Description: Alternate proof of vtxdusgr0edgnel 27555, not based on vtxduhgr0edgnel 27554. A vertex in a simple graph has degree 0 if there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vtxdushgrfvedg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdushgrfvedg.e | ⊢ 𝐸 = (Edg‘𝐺) |
vtxdushgrfvedg.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
vtxdusgr0edgnelALT | ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdushgrfvedg.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | vtxdushgrfvedg.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | vtxdushgrfvedg.d | . . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
4 | 1, 2, 3 | vtxdusgrfvedg 27551 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒})) |
5 | 4 | eqeq1d 2736 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) = 0)) |
6 | fvex 6719 | . . . . 5 ⊢ (Edg‘𝐺) ∈ V | |
7 | 2, 6 | eqeltri 2830 | . . . 4 ⊢ 𝐸 ∈ V |
8 | 7 | rabex 5214 | . . 3 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} ∈ V |
9 | hasheq0 13913 | . . 3 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} ∈ V → ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) = 0 ↔ {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} = ∅)) | |
10 | 8, 9 | mp1i 13 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) = 0 ↔ {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} = ∅)) |
11 | rabeq0 4289 | . . 3 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} = ∅ ↔ ∀𝑒 ∈ 𝐸 ¬ 𝑈 ∈ 𝑒) | |
12 | ralnex 3151 | . . . 4 ⊢ (∀𝑒 ∈ 𝐸 ¬ 𝑈 ∈ 𝑒 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒) | |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (∀𝑒 ∈ 𝐸 ¬ 𝑈 ∈ 𝑒 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒)) |
14 | 11, 13 | syl5bb 286 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ({𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} = ∅ ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒)) |
15 | 5, 10, 14 | 3bitrd 308 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ∃wrex 3055 {crab 3058 Vcvv 3401 ∅c0 4227 ‘cfv 6369 0cc0 10712 ♯chash 13879 Vtxcvtx 27059 Edgcedg 27110 USGraphcusgr 27212 VtxDegcvtxdg 27525 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-xadd 12688 df-fz 13079 df-hash 13880 df-edg 27111 df-uhgr 27121 df-ushgr 27122 df-upgr 27145 df-umgr 27146 df-uspgr 27213 df-usgr 27214 df-vtxdg 27526 |
This theorem is referenced by: (None) |
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