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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 30347 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 | ⊢ 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 |
---|---|
frgrwopregasn | ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐴 = {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 30344 | . . 3 ⊢ (𝐺 ∈ FriendGraph → ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ 𝐸) |
7 | snidg 4665 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋}) | |
8 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → 𝑋 ∈ {𝑋}) |
9 | eleq2 2828 | . . . . . . 7 ⊢ (𝐴 = {𝑋} → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ {𝑋})) | |
10 | 9 | adantl 481 | . . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ {𝑋})) |
11 | 8, 10 | mpbird 257 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → 𝑋 ∈ 𝐴) |
12 | preq1 4738 | . . . . . . . 8 ⊢ (𝑣 = 𝑋 → {𝑣, 𝑤} = {𝑋, 𝑤}) | |
13 | 12 | eleq1d 2824 | . . . . . . 7 ⊢ (𝑣 = 𝑋 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝑋, 𝑤} ∈ 𝐸)) |
14 | 13 | ralbidv 3176 | . . . . . 6 ⊢ (𝑣 = 𝑋 → (∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ 𝐸 ↔ ∀𝑤 ∈ 𝐵 {𝑋, 𝑤} ∈ 𝐸)) |
15 | 14 | rspcv 3618 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → (∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ 𝐸 → ∀𝑤 ∈ 𝐵 {𝑋, 𝑤} ∈ 𝐸)) |
16 | 11, 15 | syl 17 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → (∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ 𝐸 → ∀𝑤 ∈ 𝐵 {𝑋, 𝑤} ∈ 𝐸)) |
17 | difeq2 4130 | . . . . . . 7 ⊢ (𝐴 = {𝑋} → (𝑉 ∖ 𝐴) = (𝑉 ∖ {𝑋})) | |
18 | 4, 17 | eqtrid 2787 | . . . . . 6 ⊢ (𝐴 = {𝑋} → 𝐵 = (𝑉 ∖ {𝑋})) |
19 | 18 | adantl 481 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → 𝐵 = (𝑉 ∖ {𝑋})) |
20 | 19 | raleqdv 3324 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → (∀𝑤 ∈ 𝐵 {𝑋, 𝑤} ∈ 𝐸 ↔ ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)) |
21 | 16, 20 | sylibd 239 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → (∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐵 {𝑣, 𝑤} ∈ 𝐸 → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)) |
22 | 6, 21 | syl5com 31 | . 2 ⊢ (𝐺 ∈ FriendGraph → ((𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)) |
23 | 22 | 3impib 1115 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∖ cdif 3960 {csn 4631 {cpr 4633 ‘cfv 6563 Vtxcvtx 29028 Edgcedg 29079 VtxDegcvtxdg 29498 FriendGraph cfrgr 30287 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-xadd 13153 df-fz 13545 df-hash 14367 df-edg 29080 df-uhgr 29090 df-ushgr 29091 df-upgr 29114 df-umgr 29115 df-uspgr 29182 df-usgr 29183 df-nbgr 29365 df-vtxdg 29499 df-frgr 30288 |
This theorem is referenced by: frgrwopreg1 30347 |
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