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Mirrors > Home > MPE Home > Th. List > frgrwopreglem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for frgrwopreg 28105. If the set 𝐴 of vertices of degree 𝐾 is not empty in a friendship graph with at least two vertices, then 𝐾 must be greater than 1 . This is only an observation, which is not required for the proof the friendship theorem. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 2-Jan-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) |
Ref | Expression |
---|---|
frgrwopreglem2 | ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉) ∧ 𝐴 ≠ ∅) → 2 ≤ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4313 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | frgrwopreg.a | . . . . . 6 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} | |
3 | 2 | rabeq2i 3490 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾)) |
4 | frgrwopreg.v | . . . . . . . . . . 11 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | vdgfrgrgt2 28080 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑥 ∈ 𝑉) → (1 < (♯‘𝑉) → 2 ≤ ((VtxDeg‘𝐺)‘𝑥))) |
6 | 5 | imp 409 | . . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑥 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → 2 ≤ ((VtxDeg‘𝐺)‘𝑥)) |
7 | breq2 5073 | . . . . . . . . . . 11 ⊢ (𝐾 = (𝐷‘𝑥) → (2 ≤ 𝐾 ↔ 2 ≤ (𝐷‘𝑥))) | |
8 | frgrwopreg.d | . . . . . . . . . . . . 13 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
9 | 8 | fveq1i 6674 | . . . . . . . . . . . 12 ⊢ (𝐷‘𝑥) = ((VtxDeg‘𝐺)‘𝑥) |
10 | 9 | breq2i 5077 | . . . . . . . . . . 11 ⊢ (2 ≤ (𝐷‘𝑥) ↔ 2 ≤ ((VtxDeg‘𝐺)‘𝑥)) |
11 | 7, 10 | syl6bb 289 | . . . . . . . . . 10 ⊢ (𝐾 = (𝐷‘𝑥) → (2 ≤ 𝐾 ↔ 2 ≤ ((VtxDeg‘𝐺)‘𝑥))) |
12 | 11 | eqcoms 2832 | . . . . . . . . 9 ⊢ ((𝐷‘𝑥) = 𝐾 → (2 ≤ 𝐾 ↔ 2 ≤ ((VtxDeg‘𝐺)‘𝑥))) |
13 | 6, 12 | syl5ibrcom 249 | . . . . . . . 8 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑥 ∈ 𝑉) ∧ 1 < (♯‘𝑉)) → ((𝐷‘𝑥) = 𝐾 → 2 ≤ 𝐾)) |
14 | 13 | exp31 422 | . . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → (𝑥 ∈ 𝑉 → (1 < (♯‘𝑉) → ((𝐷‘𝑥) = 𝐾 → 2 ≤ 𝐾)))) |
15 | 14 | com14 96 | . . . . . 6 ⊢ ((𝐷‘𝑥) = 𝐾 → (𝑥 ∈ 𝑉 → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → 2 ≤ 𝐾)))) |
16 | 15 | impcom 410 | . . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐷‘𝑥) = 𝐾) → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → 2 ≤ 𝐾))) |
17 | 3, 16 | sylbi 219 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → 2 ≤ 𝐾))) |
18 | 17 | exlimiv 1930 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → 2 ≤ 𝐾))) |
19 | 1, 18 | sylbi 219 | . 2 ⊢ (𝐴 ≠ ∅ → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → 2 ≤ 𝐾))) |
20 | 19 | 3imp31 1108 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉) ∧ 𝐴 ≠ ∅) → 2 ≤ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3019 {crab 3145 ∖ cdif 3936 ∅c0 4294 class class class wbr 5069 ‘cfv 6358 1c1 10541 < clt 10678 ≤ cle 10679 2c2 11695 ♯chash 13693 Vtxcvtx 26784 VtxDegcvtxdg 27250 FriendGraph cfrgr 28040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-xadd 12511 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13926 df-s1 13953 df-s2 14213 df-s3 14214 df-edg 26836 df-uhgr 26846 df-upgr 26870 df-umgr 26871 df-uspgr 26938 df-usgr 26939 df-vtxdg 27251 df-wlks 27384 df-wlkson 27385 df-trls 27477 df-trlson 27478 df-pths 27500 df-spths 27501 df-pthson 27502 df-spthson 27503 df-conngr 27969 df-frgr 28041 |
This theorem is referenced by: (None) |
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