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Mirrors > Home > MPE Home > Th. List > vtxdusgr0edgnelALT | Structured version Visualization version GIF version |
Description: Alternate proof of vtxdusgr0edgnel 27280, not based on vtxduhgr0edgnel 27279. 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 27276 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (𝐷‘𝑈) = (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒})) |
5 | 4 | eqeq1d 2826 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐷‘𝑈) = 0 ↔ (♯‘{𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒}) = 0)) |
6 | fvex 6686 | . . . . 5 ⊢ (Edg‘𝐺) ∈ V | |
7 | 2, 6 | eqeltri 2912 | . . . 4 ⊢ 𝐸 ∈ V |
8 | 7 | rabex 5238 | . . 3 ⊢ {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} ∈ V |
9 | hasheq0 13727 | . . 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 4341 | . . 3 ⊢ ({𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} = ∅ ↔ ∀𝑒 ∈ 𝐸 ¬ 𝑈 ∈ 𝑒) | |
12 | ralnex 3239 | . . . 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 285 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ({𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} = ∅ ↔ ¬ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒)) |
15 | 5, 10, 14 | 3bitrd 307 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 {crab 3145 Vcvv 3497 ∅c0 4294 ‘cfv 6358 0cc0 10540 ♯chash 13693 Vtxcvtx 26784 Edgcedg 26835 USGraphcusgr 26937 VtxDegcvtxdg 27250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-xadd 12511 df-fz 12896 df-hash 13694 df-edg 26836 df-uhgr 26846 df-ushgr 26847 df-upgr 26870 df-umgr 26871 df-uspgr 26938 df-usgr 26939 df-vtxdg 27251 |
This theorem is referenced by: (None) |
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