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Mirrors > Home > MPE Home > Th. List > nbgrassovt | Structured version Visualization version GIF version |
Description: The neighbors of a vertex are a subset of the other vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) |
Ref | Expression |
---|---|
nbgrassovt | ⊢ (𝑉 USGrph 𝐸 → (𝑁 ∈ 𝑉 → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgranself 25963 | . . . 4 ⊢ (𝑉 USGrph 𝐸 → ∀𝑛 ∈ 𝑉 𝑛 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑛)) | |
2 | id 22 | . . . . . 6 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁) | |
3 | oveq2 6557 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (〈𝑉, 𝐸〉 Neighbors 𝑛) = (〈𝑉, 𝐸〉 Neighbors 𝑁)) | |
4 | 2, 3 | neleq12d 2887 | . . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑛) ↔ 𝑁 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁))) |
5 | 4 | rspcv 3278 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (∀𝑛 ∈ 𝑉 𝑛 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑛) → 𝑁 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁))) |
6 | 1, 5 | mpan9 485 | . . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁)) |
7 | nbgrassvt 25962 | . . . 4 ⊢ (𝑉 USGrph 𝐸 → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉) |
9 | df-nel 2783 | . . . 4 ⊢ (𝑁 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁) ↔ ¬ 𝑁 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁)) | |
10 | difsn 4269 | . . . . 5 ⊢ (¬ 𝑁 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁) → ((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) = (〈𝑉, 𝐸〉 Neighbors 𝑁)) | |
11 | simpl 472 | . . . . . . 7 ⊢ ((((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) = (〈𝑉, 𝐸〉 Neighbors 𝑁) ∧ (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉) → ((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) = (〈𝑉, 𝐸〉 Neighbors 𝑁)) | |
12 | simpr 476 | . . . . . . . 8 ⊢ ((((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) = (〈𝑉, 𝐸〉 Neighbors 𝑁) ∧ (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉) | |
13 | 12 | ssdifd 3708 | . . . . . . 7 ⊢ ((((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) = (〈𝑉, 𝐸〉 Neighbors 𝑁) ∧ (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉) → ((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) ⊆ (𝑉 ∖ {𝑁})) |
14 | 11, 13 | eqsstr3d 3603 | . . . . . 6 ⊢ ((((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) = (〈𝑉, 𝐸〉 Neighbors 𝑁) ∧ (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁})) |
15 | 14 | ex 449 | . . . . 5 ⊢ (((〈𝑉, 𝐸〉 Neighbors 𝑁) ∖ {𝑁}) = (〈𝑉, 𝐸〉 Neighbors 𝑁) → ((〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉 → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}))) |
16 | 10, 15 | syl 17 | . . . 4 ⊢ (¬ 𝑁 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁) → ((〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉 → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}))) |
17 | 9, 16 | sylbi 206 | . . 3 ⊢ (𝑁 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁) → ((〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ 𝑉 → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}))) |
18 | 6, 8, 17 | sylc 63 | . 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁})) |
19 | 18 | ex 449 | 1 ⊢ (𝑉 USGrph 𝐸 → (𝑁 ∈ 𝑉 → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 ∀wral 2896 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 〈cop 4131 class class class wbr 4583 (class class class)co 6549 USGrph cusg 25859 Neighbors cnbgra 25946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-usgra 25862 df-nbgra 25949 |
This theorem is referenced by: nbgranself2 25965 |
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