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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 | β’ π = (VtxβπΊ) |
frgrwopreg.d | β’ π· = (VtxDegβπΊ) |
frgrwopreg.a | β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} |
frgrwopreg.b | β’ π΅ = (π β π΄) |
frgrwopreg.e | β’ πΈ = (EdgβπΊ) |
Ref | Expression |
---|---|
frgrwopreg2 | β’ ((πΊ β FriendGraph β§ (β―βπ΅) = 1) β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.v | . . . . . 6 β’ π = (VtxβπΊ) | |
2 | frgrwopreg.d | . . . . . 6 β’ π· = (VtxDegβπΊ) | |
3 | frgrwopreg.a | . . . . . 6 β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} | |
4 | frgrwopreg.b | . . . . . 6 β’ π΅ = (π β π΄) | |
5 | 1, 2, 3, 4 | frgrwopreglem1 28717 | . . . . 5 β’ (π΄ β V β§ π΅ β V) |
6 | 5 | simpri 487 | . . . 4 β’ π΅ β V |
7 | hash1snb 14175 | . . . 4 β’ (π΅ β V β ((β―βπ΅) = 1 β βπ£ π΅ = {π£})) | |
8 | 6, 7 | ax-mp 5 | . . 3 β’ ((β―βπ΅) = 1 β βπ£ π΅ = {π£}) |
9 | exsnrex 4620 | . . . . 5 β’ (βπ£ π΅ = {π£} β βπ£ β π΅ π΅ = {π£}) | |
10 | difss 4072 | . . . . . . . 8 β’ (π β π΄) β π | |
11 | 4, 10 | eqsstri 3960 | . . . . . . 7 β’ π΅ β π |
12 | ssrexv 3993 | . . . . . . 7 β’ (π΅ β π β (βπ£ β π΅ π΅ = {π£} β βπ£ β π π΅ = {π£})) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 β’ (βπ£ β π΅ π΅ = {π£} β βπ£ β π π΅ = {π£}) |
14 | frgrwopreg.e | . . . . . . . . 9 β’ πΈ = (EdgβπΊ) | |
15 | 1, 2, 3, 4, 14 | frgrwopregbsn 28722 | . . . . . . . 8 β’ ((πΊ β FriendGraph β§ π£ β π β§ π΅ = {π£}) β βπ€ β (π β {π£}){π£, π€} β πΈ) |
16 | 15 | 3expia 1121 | . . . . . . 7 β’ ((πΊ β FriendGraph β§ π£ β π) β (π΅ = {π£} β βπ€ β (π β {π£}){π£, π€} β πΈ)) |
17 | 16 | reximdva 3162 | . . . . . 6 β’ (πΊ β FriendGraph β (βπ£ β π π΅ = {π£} β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
18 | 13, 17 | syl5com 31 | . . . . 5 β’ (βπ£ β π΅ π΅ = {π£} β (πΊ β FriendGraph β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
19 | 9, 18 | sylbi 216 | . . . 4 β’ (βπ£ π΅ = {π£} β (πΊ β FriendGraph β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
20 | 19 | com12 32 | . . 3 β’ (πΊ β FriendGraph β (βπ£ π΅ = {π£} β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
21 | 8, 20 | syl5bi 243 | . 2 β’ (πΊ β FriendGraph β ((β―βπ΅) = 1 β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) |
22 | 21 | imp 408 | 1 β’ ((πΊ β FriendGraph β§ (β―βπ΅) = 1) β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1539 βwex 1779 β wcel 2104 βwral 3062 βwrex 3071 {crab 3284 Vcvv 3437 β cdif 3889 β wss 3892 {csn 4565 {cpr 4567 βcfv 6454 1c1 10914 β―chash 14086 Vtxcvtx 27407 Edgcedg 27458 VtxDegcvtxdg 27873 FriendGraph cfrgr 28663 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-2o 8325 df-oadd 8328 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-dju 9699 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-nn 12016 df-2 12078 df-n0 12276 df-xnn0 12348 df-z 12362 df-uz 12625 df-xadd 12891 df-fz 13282 df-hash 14087 df-edg 27459 df-uhgr 27469 df-ushgr 27470 df-upgr 27493 df-umgr 27494 df-uspgr 27561 df-usgr 27562 df-nbgr 27741 df-vtxdg 27874 df-frgr 28664 |
This theorem is referenced by: frgrregorufr0 28729 |
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