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Mirrors > Home > MPE Home > Th. List > frgrncvvdeqlem7 | Structured version Visualization version GIF version |
Description: Lemma 7 for frgrncvvdeq 28067. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) |
Ref | Expression |
---|---|
frgrncvvdeq.v1 | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrncvvdeq.e | ⊢ 𝐸 = (Edg‘𝐺) |
frgrncvvdeq.nx | ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) |
frgrncvvdeq.ny | ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) |
frgrncvvdeq.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
frgrncvvdeq.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
frgrncvvdeq.ne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
frgrncvvdeq.xy | ⊢ (𝜑 → 𝑌 ∉ 𝐷) |
frgrncvvdeq.f | ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) |
frgrncvvdeq.a | ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) |
Ref | Expression |
---|---|
frgrncvvdeqlem7 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrncvvdeq.v1 | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frgrncvvdeq.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | frgrncvvdeq.nx | . . . 4 ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) | |
4 | frgrncvvdeq.ny | . . . 4 ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) | |
5 | frgrncvvdeq.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
6 | frgrncvvdeq.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | frgrncvvdeq.ne | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
8 | frgrncvvdeq.xy | . . . 4 ⊢ (𝜑 → 𝑌 ∉ 𝐷) | |
9 | frgrncvvdeq.f | . . . 4 ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) | |
10 | frgrncvvdeq.a | . . . 4 ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | frgrncvvdeqlem5 28061 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) |
12 | fvex 6664 | . . . . 5 ⊢ (𝐴‘𝑥) ∈ V | |
13 | 12 | snid 4582 | . . . 4 ⊢ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)} |
14 | eleq2 2899 | . . . . . 6 ⊢ ({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝐴‘𝑥) ∈ {(𝐴‘𝑥)} ↔ (𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁))) | |
15 | 14 | biimpa 479 | . . . . 5 ⊢ (({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → (𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) |
16 | elin 4152 | . . . . . 6 ⊢ ((𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ↔ ((𝐴‘𝑥) ∈ (𝐺 NeighbVtx 𝑥) ∧ (𝐴‘𝑥) ∈ 𝑁)) | |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | frgrncvvdeqlem1 28057 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∉ 𝑁) |
18 | df-nel 3119 | . . . . . . . . . 10 ⊢ (𝑋 ∉ 𝑁 ↔ ¬ 𝑋 ∈ 𝑁) | |
19 | nelelne 3112 | . . . . . . . . . 10 ⊢ (¬ 𝑋 ∈ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) | |
20 | 18, 19 | sylbi 219 | . . . . . . . . 9 ⊢ (𝑋 ∉ 𝑁 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) |
21 | 17, 20 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) |
22 | 21 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐴‘𝑥) ∈ 𝑁 → (𝐴‘𝑥) ≠ 𝑋)) |
23 | 22 | com12 32 | . . . . . 6 ⊢ ((𝐴‘𝑥) ∈ 𝑁 → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
24 | 16, 23 | simplbiim 507 | . . . . 5 ⊢ ((𝐴‘𝑥) ∈ ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
25 | 15, 24 | syl 17 | . . . 4 ⊢ (({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) ∧ (𝐴‘𝑥) ∈ {(𝐴‘𝑥)}) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
26 | 13, 25 | mpan2 689 | . . 3 ⊢ ({(𝐴‘𝑥)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁) → ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋)) |
27 | 11, 26 | mpcom 38 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐴‘𝑥) ≠ 𝑋) |
28 | 27 | ralrimiva 3177 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) ≠ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3011 ∉ wnel 3118 ∀wral 3133 ∩ cin 3918 {csn 4548 {cpr 4550 ↦ cmpt 5127 ‘cfv 6336 ℩crio 7094 (class class class)co 7137 Vtxcvtx 26762 Edgcedg 26813 NeighbVtx cnbgr 27095 FriendGraph cfrgr 28016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-2o 8084 df-oadd 8087 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-dju 9311 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-n0 11880 df-xnn0 11950 df-z 11964 df-uz 12226 df-fz 12878 df-hash 13676 df-edg 26814 df-upgr 26848 df-umgr 26849 df-usgr 26917 df-nbgr 27096 df-frgr 28017 |
This theorem is referenced by: (None) |
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