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Mirrors > Home > MPE Home > Th. List > frgrwopregasn | 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". This version of frgrwopreg1 30115 is stricter (claiming that the singleton itself is a universal friend instead of claiming the existence of a universal friend only) and therefore closer to Huneke's statement. This strict variant, however, is not required for the proof of the friendship theorem. (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised 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 |
---|---|
frgrwopregasn | β’ ((πΊ β FriendGraph β§ π β π β§ π΄ = {π}) β βπ€ β (π β {π}){π, π€} β πΈ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.v | . . . 4 β’ π = (VtxβπΊ) | |
2 | frgrwopreg.d | . . . 4 β’ π· = (VtxDegβπΊ) | |
3 | frgrwopreg.a | . . . 4 β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} | |
4 | frgrwopreg.b | . . . 4 β’ π΅ = (π β π΄) | |
5 | frgrwopreg.e | . . . 4 β’ πΈ = (EdgβπΊ) | |
6 | 1, 2, 3, 4, 5 | frgrwopreglem4 30112 | . . 3 β’ (πΊ β FriendGraph β βπ£ β π΄ βπ€ β π΅ {π£, π€} β πΈ) |
7 | snidg 4658 | . . . . . . 7 β’ (π β π β π β {π}) | |
8 | 7 | adantr 480 | . . . . . 6 β’ ((π β π β§ π΄ = {π}) β π β {π}) |
9 | eleq2 2817 | . . . . . . 7 β’ (π΄ = {π} β (π β π΄ β π β {π})) | |
10 | 9 | adantl 481 | . . . . . 6 β’ ((π β π β§ π΄ = {π}) β (π β π΄ β π β {π})) |
11 | 8, 10 | mpbird 257 | . . . . 5 β’ ((π β π β§ π΄ = {π}) β π β π΄) |
12 | preq1 4733 | . . . . . . . 8 β’ (π£ = π β {π£, π€} = {π, π€}) | |
13 | 12 | eleq1d 2813 | . . . . . . 7 β’ (π£ = π β ({π£, π€} β πΈ β {π, π€} β πΈ)) |
14 | 13 | ralbidv 3172 | . . . . . 6 β’ (π£ = π β (βπ€ β π΅ {π£, π€} β πΈ β βπ€ β π΅ {π, π€} β πΈ)) |
15 | 14 | rspcv 3603 | . . . . 5 β’ (π β π΄ β (βπ£ β π΄ βπ€ β π΅ {π£, π€} β πΈ β βπ€ β π΅ {π, π€} β πΈ)) |
16 | 11, 15 | syl 17 | . . . 4 β’ ((π β π β§ π΄ = {π}) β (βπ£ β π΄ βπ€ β π΅ {π£, π€} β πΈ β βπ€ β π΅ {π, π€} β πΈ)) |
17 | difeq2 4112 | . . . . . . 7 β’ (π΄ = {π} β (π β π΄) = (π β {π})) | |
18 | 4, 17 | eqtrid 2779 | . . . . . 6 β’ (π΄ = {π} β π΅ = (π β {π})) |
19 | 18 | adantl 481 | . . . . 5 β’ ((π β π β§ π΄ = {π}) β π΅ = (π β {π})) |
20 | 19 | raleqdv 3320 | . . . 4 β’ ((π β π β§ π΄ = {π}) β (βπ€ β π΅ {π, π€} β πΈ β βπ€ β (π β {π}){π, π€} β πΈ)) |
21 | 16, 20 | sylibd 238 | . . 3 β’ ((π β π β§ π΄ = {π}) β (βπ£ β π΄ βπ€ β π΅ {π£, π€} β πΈ β βπ€ β (π β {π}){π, π€} β πΈ)) |
22 | 6, 21 | syl5com 31 | . 2 β’ (πΊ β FriendGraph β ((π β π β§ π΄ = {π}) β βπ€ β (π β {π}){π, π€} β πΈ)) |
23 | 22 | 3impib 1114 | 1 β’ ((πΊ β FriendGraph β§ π β π β§ π΄ = {π}) β 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3056 {crab 3427 β cdif 3941 {csn 4624 {cpr 4626 βcfv 6542 Vtxcvtx 28796 Edgcedg 28847 VtxDegcvtxdg 29266 FriendGraph cfrgr 30055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-xadd 13117 df-fz 13509 df-hash 14314 df-edg 28848 df-uhgr 28858 df-ushgr 28859 df-upgr 28882 df-umgr 28883 df-uspgr 28950 df-usgr 28951 df-nbgr 29133 df-vtxdg 29267 df-frgr 30056 |
This theorem is referenced by: frgrwopreg1 30115 |
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