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Mirrors > Home > MPE Home > Th. List > frgrwopreg2 | Structured version Visualization version GIF version |
Description: According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Proof shortened by AV, 4-Feb-2022.) |
Ref | Expression |
---|---|
frgrwopreg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrwopreg.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
frgrwopreg.a | ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} |
frgrwopreg.b | ⊢ 𝐵 = (𝑉 ∖ 𝐴) |
frgrwopreg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
frgrwopreg2 | ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘𝐵) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frgrwopreg.d | . . . . . 6 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
3 | frgrwopreg.a | . . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
4 | frgrwopreg.b | . . . . . 6 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
5 | 1, 2, 3, 4 | frgrwopreglem1 28018 | . . . . 5 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
6 | 5 | simpri 486 | . . . 4 ⊢ 𝐵 ∈ V |
7 | hash1snb 13768 | . . . 4 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ ∃𝑣 𝐵 = {𝑣})) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ ((♯‘𝐵) = 1 ↔ ∃𝑣 𝐵 = {𝑣}) |
9 | exsnrex 4610 | . . . . 5 ⊢ (∃𝑣 𝐵 = {𝑣} ↔ ∃𝑣 ∈ 𝐵 𝐵 = {𝑣}) | |
10 | difss 4105 | . . . . . . . 8 ⊢ (𝑉 ∖ 𝐴) ⊆ 𝑉 | |
11 | 4, 10 | eqsstri 3998 | . . . . . . 7 ⊢ 𝐵 ⊆ 𝑉 |
12 | ssrexv 4031 | . . . . . . 7 ⊢ (𝐵 ⊆ 𝑉 → (∃𝑣 ∈ 𝐵 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐵 = {𝑣})) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ (∃𝑣 ∈ 𝐵 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐵 = {𝑣}) |
14 | frgrwopreg.e | . . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺) | |
15 | 1, 2, 3, 4, 14 | frgrwopregbsn 28023 | . . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉 ∧ 𝐵 = {𝑣}) → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) |
16 | 15 | 3expia 1113 | . . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) → (𝐵 = {𝑣} → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
17 | 16 | reximdva 3271 | . . . . . 6 ⊢ (𝐺 ∈ FriendGraph → (∃𝑣 ∈ 𝑉 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
18 | 13, 17 | syl5com 31 | . . . . 5 ⊢ (∃𝑣 ∈ 𝐵 𝐵 = {𝑣} → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
19 | 9, 18 | sylbi 218 | . . . 4 ⊢ (∃𝑣 𝐵 = {𝑣} → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
20 | 19 | com12 32 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (∃𝑣 𝐵 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
21 | 8, 20 | syl5bi 243 | . 2 ⊢ (𝐺 ∈ FriendGraph → ((♯‘𝐵) = 1 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
22 | 21 | imp 407 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘𝐵) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 {crab 3139 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 {cpr 4559 ‘cfv 6348 1c1 10526 ♯chash 13678 Vtxcvtx 26708 Edgcedg 26759 VtxDegcvtxdg 27174 FriendGraph cfrgr 27964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12881 df-hash 13679 df-edg 26760 df-uhgr 26770 df-ushgr 26771 df-upgr 26794 df-umgr 26795 df-uspgr 26862 df-usgr 26863 df-nbgr 27042 df-vtxdg 27175 df-frgr 27965 |
This theorem is referenced by: frgrregorufr0 28030 |
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