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Mirrors > Home > MPE Home > Th. List > frgrwopregasn | Structured version Visualization version Unicode 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 27182 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 | Vtx |
frgrwopreg.d | VtxDeg |
frgrwopreg.a | |
frgrwopreg.b | |
frgrwopreg.e | Edg |
Ref | Expression |
---|---|
frgrwopregasn | FriendGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.v | . . . 4 Vtx | |
2 | frgrwopreg.d | . . . 4 VtxDeg | |
3 | frgrwopreg.a | . . . 4 | |
4 | frgrwopreg.b | . . . 4 | |
5 | frgrwopreg.e | . . . 4 Edg | |
6 | 1, 2, 3, 4, 5 | frgrwopreglem4 27179 | . . 3 FriendGraph |
7 | snidg 4206 | . . . . . . 7 | |
8 | 7 | adantr 481 | . . . . . 6 |
9 | eleq2 2690 | . . . . . . 7 | |
10 | 9 | adantl 482 | . . . . . 6 |
11 | 8, 10 | mpbird 247 | . . . . 5 |
12 | preq1 4268 | . . . . . . . 8 | |
13 | 12 | eleq1d 2686 | . . . . . . 7 |
14 | 13 | ralbidv 2986 | . . . . . 6 |
15 | 14 | rspcv 3305 | . . . . 5 |
16 | 11, 15 | syl 17 | . . . 4 |
17 | difeq2 3722 | . . . . . . 7 | |
18 | 4, 17 | syl5eq 2668 | . . . . . 6 |
19 | 18 | adantl 482 | . . . . 5 |
20 | 19 | raleqdv 3144 | . . . 4 |
21 | 16, 20 | sylibd 229 | . . 3 |
22 | 6, 21 | syl5com 31 | . 2 FriendGraph |
23 | 22 | 3impib 1262 | 1 FriendGraph |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 crab 2916 cdif 3571 csn 4177 cpr 4179 cfv 5888 Vtxcvtx 25874 Edgcedg 25939 VtxDegcvtxdg 26361 FriendGraph cfrgr 27120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-xadd 11947 df-fz 12327 df-hash 13118 df-edg 25940 df-uhgr 25953 df-ushgr 25954 df-upgr 25977 df-umgr 25978 df-uspgr 26045 df-usgr 26046 df-nbgr 26228 df-vtxdg 26362 df-frgr 27121 |
This theorem is referenced by: frgrwopreg1 27182 |
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