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| Description: Lemma 4 for frgrwopreg 30343. In a friendship graph each vertex with degree 𝐾 is connected with any vertex with degree other than 𝐾. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened 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 | 
|---|---|
| frgrwopreglem4 | ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ 𝐸) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl 482 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝐺 ∈ FriendGraph ) | |
| 2 | elrabi 3686 | . . . . . 6 ⊢ (𝑎 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑎 ∈ 𝑉) | |
| 3 | frgrwopreg.a | . . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
| 4 | 2, 3 | eleq2s 2858 | . . . . 5 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉) | 
| 5 | eldifi 4130 | . . . . . 6 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) → 𝑏 ∈ 𝑉) | |
| 6 | frgrwopreg.b | . . . . . 6 ⊢ 𝐵 = (𝑉 ∖ 𝐴) | |
| 7 | 5, 6 | eleq2s 2858 | . . . . 5 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑉) | 
| 8 | 4, 7 | anim12i 613 | . . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) | 
| 9 | 8 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) | 
| 10 | frgrwopreg.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 11 | frgrwopreg.d | . . . . 5 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
| 12 | 10, 11, 3, 6 | frgrwopreglem3 30334 | . . . 4 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐷‘𝑎) ≠ (𝐷‘𝑏)) | 
| 13 | 12 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → (𝐷‘𝑎) ≠ (𝐷‘𝑏)) | 
| 14 | frgrwopreg.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
| 15 | 10, 11, 14 | frgrwopreglem4a 30330 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏)) → {𝑎, 𝑏} ∈ 𝐸) | 
| 16 | 1, 9, 13, 15 | syl3anc 1372 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → {𝑎, 𝑏} ∈ 𝐸) | 
| 17 | 16 | ralrimivva 3201 | 1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ 𝐸) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 {crab 3435 ∖ cdif 3947 {cpr 4627 ‘cfv 6560 Vtxcvtx 29014 Edgcedg 29065 VtxDegcvtxdg 29484 FriendGraph cfrgr 30278 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-xadd 13156 df-fz 13549 df-hash 14371 df-edg 29066 df-uhgr 29076 df-ushgr 29077 df-upgr 29100 df-umgr 29101 df-uspgr 29168 df-usgr 29169 df-nbgr 29351 df-vtxdg 29485 df-frgr 30279 | 
| This theorem is referenced by: frgrwopregasn 30336 frgrwopregbsn 30337 frgrwopreglem5ALT 30342 | 
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