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Mirrors > Home > MPE Home > Th. List > frgrwopregbsn | 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". This version of frgrwopreg2 28092 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 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 |
---|---|
frgrwopregbsn | ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrwopreg.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frgrwopreg.d | . . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
3 | frgrwopreg.a | . . . 4 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
4 | frgrwopreg.b | . . . 4 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
5 | frgrwopreg.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
6 | 1, 2, 3, 4, 5 | frgrwopreglem4 28088 | . . 3 ⊢ (𝐺 ∈ FriendGraph → ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 {𝑤, 𝑣} ∈ 𝐸) |
7 | ralcom 3354 | . . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 {𝑤, 𝑣} ∈ 𝐸 ↔ ∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ 𝐸) | |
8 | snidg 4592 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋}) | |
9 | 8 | adantr 483 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → 𝑋 ∈ {𝑋}) |
10 | eleq2 2901 | . . . . . . . 8 ⊢ (𝐵 = {𝑋} → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {𝑋})) | |
11 | 10 | adantl 484 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {𝑋})) |
12 | 9, 11 | mpbird 259 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → 𝑋 ∈ 𝐵) |
13 | preq2 4663 | . . . . . . . . . 10 ⊢ (𝑣 = 𝑋 → {𝑤, 𝑣} = {𝑤, 𝑋}) | |
14 | prcom 4661 | . . . . . . . . . 10 ⊢ {𝑤, 𝑋} = {𝑋, 𝑤} | |
15 | 13, 14 | syl6eq 2872 | . . . . . . . . 9 ⊢ (𝑣 = 𝑋 → {𝑤, 𝑣} = {𝑋, 𝑤}) |
16 | 15 | eleq1d 2897 | . . . . . . . 8 ⊢ (𝑣 = 𝑋 → ({𝑤, 𝑣} ∈ 𝐸 ↔ {𝑋, 𝑤} ∈ 𝐸)) |
17 | 16 | ralbidv 3197 | . . . . . . 7 ⊢ (𝑣 = 𝑋 → (∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ 𝐸 ↔ ∀𝑤 ∈ 𝐴 {𝑋, 𝑤} ∈ 𝐸)) |
18 | 17 | rspcv 3617 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ 𝐸 → ∀𝑤 ∈ 𝐴 {𝑋, 𝑤} ∈ 𝐸)) |
19 | 12, 18 | syl 17 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → (∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ 𝐸 → ∀𝑤 ∈ 𝐴 {𝑋, 𝑤} ∈ 𝐸)) |
20 | 3 | ssrab3 4056 | . . . . . . . 8 ⊢ 𝐴 ⊆ 𝑉 |
21 | ssdifim 4238 | . . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵)) | |
22 | 20, 4, 21 | mp2an 690 | . . . . . . 7 ⊢ 𝐴 = (𝑉 ∖ 𝐵) |
23 | difeq2 4092 | . . . . . . . 8 ⊢ (𝐵 = {𝑋} → (𝑉 ∖ 𝐵) = (𝑉 ∖ {𝑋})) | |
24 | 23 | adantl 484 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → (𝑉 ∖ 𝐵) = (𝑉 ∖ {𝑋})) |
25 | 22, 24 | syl5eq 2868 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → 𝐴 = (𝑉 ∖ {𝑋})) |
26 | 25 | raleqdv 3415 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → (∀𝑤 ∈ 𝐴 {𝑋, 𝑤} ∈ 𝐸 ↔ ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)) |
27 | 19, 26 | sylibd 241 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → (∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐴 {𝑤, 𝑣} ∈ 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)) |
28 | 7, 27 | syl5bi 244 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 {𝑤, 𝑣} ∈ 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)) |
29 | 6, 28 | syl5com 31 | . 2 ⊢ (𝐺 ∈ FriendGraph → ((𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)) |
30 | 29 | 3impib 1112 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ∖ cdif 3932 ⊆ wss 3935 {csn 4560 {cpr 4562 ‘cfv 6349 Vtxcvtx 26775 Edgcedg 26826 VtxDegcvtxdg 27241 FriendGraph cfrgr 28031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-xadd 12502 df-fz 12887 df-hash 13685 df-edg 26827 df-uhgr 26837 df-ushgr 26838 df-upgr 26861 df-umgr 26862 df-uspgr 26929 df-usgr 26930 df-nbgr 27109 df-vtxdg 27242 df-frgr 28032 |
This theorem is referenced by: frgrwopreg2 28092 |
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