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Mirrors > Home > MPE Home > Th. List > frgrwopreg1 | Structured version Visualization version GIF version |
Description: According to statement 5 in [Huneke] p. 2: "If A ... 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 |
---|---|
frgrwopreg1 | ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘𝐴) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.a | . . . . 5 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
2 | frgrwopreg.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | fvexi 6854 | . . . . 5 ⊢ 𝑉 ∈ V |
4 | 1, 3 | rabex2 5290 | . . . 4 ⊢ 𝐴 ∈ V |
5 | hash1snb 14316 | . . . 4 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ ∃𝑣 𝐴 = {𝑣})) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ((♯‘𝐴) = 1 ↔ ∃𝑣 𝐴 = {𝑣}) |
7 | exsnrex 4640 | . . . . 5 ⊢ (∃𝑣 𝐴 = {𝑣} ↔ ∃𝑣 ∈ 𝐴 𝐴 = {𝑣}) | |
8 | 1 | ssrab3 4039 | . . . . . . 7 ⊢ 𝐴 ⊆ 𝑉 |
9 | ssrexv 4010 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝑉 → (∃𝑣 ∈ 𝐴 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐴 = {𝑣})) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (∃𝑣 ∈ 𝐴 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 𝐴 = {𝑣}) |
11 | frgrwopreg.d | . . . . . . . . 9 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
12 | frgrwopreg.b | . . . . . . . . 9 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
13 | frgrwopreg.e | . . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺) | |
14 | 2, 11, 1, 12, 13 | frgrwopregasn 29158 | . . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉 ∧ 𝐴 = {𝑣}) → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) |
15 | 14 | 3expia 1121 | . . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑣 ∈ 𝑉) → (𝐴 = {𝑣} → ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
16 | 15 | reximdva 3164 | . . . . . 6 ⊢ (𝐺 ∈ FriendGraph → (∃𝑣 ∈ 𝑉 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
17 | 10, 16 | syl5com 31 | . . . . 5 ⊢ (∃𝑣 ∈ 𝐴 𝐴 = {𝑣} → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
18 | 7, 17 | sylbi 216 | . . . 4 ⊢ (∃𝑣 𝐴 = {𝑣} → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
19 | 18 | com12 32 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (∃𝑣 𝐴 = {𝑣} → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
20 | 6, 19 | biimtrid 241 | . 2 ⊢ (𝐺 ∈ FriendGraph → ((♯‘𝐴) = 1 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
21 | 20 | imp 407 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘𝐴) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3063 ∃wrex 3072 {crab 3406 Vcvv 3444 ∖ cdif 3906 ⊆ wss 3909 {csn 4585 {cpr 4587 ‘cfv 6494 1c1 11049 ♯chash 14227 Vtxcvtx 27845 Edgcedg 27896 VtxDegcvtxdg 28311 FriendGraph cfrgr 29100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-2o 8410 df-oadd 8413 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-dju 9834 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-n0 12411 df-xnn0 12483 df-z 12497 df-uz 12761 df-xadd 13031 df-fz 13422 df-hash 14228 df-edg 27897 df-uhgr 27907 df-ushgr 27908 df-upgr 27931 df-umgr 27932 df-uspgr 27999 df-usgr 28000 df-nbgr 28179 df-vtxdg 28312 df-frgr 29101 |
This theorem is referenced by: frgrregorufr0 29166 |
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