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Mirrors > Home > MPE Home > Th. List > nbgrassvwo2 | Structured version Visualization version GIF version |
Description: The neighbors of a vertex in a graph are a subset of all vertices of the graph except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) |
Ref | Expression |
---|---|
nbgrassvwo2 | ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑀 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁)) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrassvwo 25966 | . . . 4 ⊢ (𝑉 USGrph 𝐸 → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁})) | |
2 | df-nel 2783 | . . . . . 6 ⊢ (𝑀 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁) ↔ ¬ 𝑀 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁)) | |
3 | disjsn 4192 | . . . . . 6 ⊢ (((〈𝑉, 𝐸〉 Neighbors 𝑁) ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ (〈𝑉, 𝐸〉 Neighbors 𝑁)) | |
4 | 2, 3 | sylbb2 227 | . . . . 5 ⊢ (𝑀 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁) → ((〈𝑉, 𝐸〉 Neighbors 𝑁) ∩ {𝑀}) = ∅) |
5 | reldisj 3972 | . . . . 5 ⊢ ((〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (((〈𝑉, 𝐸〉 Neighbors 𝑁) ∩ {𝑀}) = ∅ ↔ (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))) | |
6 | 4, 5 | syl5ib 233 | . . . 4 ⊢ ((〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑁}) → (𝑀 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))) |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝑉 USGrph 𝐸 → (𝑀 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀}))) |
8 | 7 | imp 444 | . 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑀 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁)) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ ((𝑉 ∖ {𝑁}) ∖ {𝑀})) |
9 | prcom 4211 | . . . 4 ⊢ {𝑀, 𝑁} = {𝑁, 𝑀} | |
10 | 9 | difeq2i 3687 | . . 3 ⊢ (𝑉 ∖ {𝑀, 𝑁}) = (𝑉 ∖ {𝑁, 𝑀}) |
11 | difpr 4275 | . . 3 ⊢ (𝑉 ∖ {𝑁, 𝑀}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀}) | |
12 | 10, 11 | eqtri 2632 | . 2 ⊢ (𝑉 ∖ {𝑀, 𝑁}) = ((𝑉 ∖ {𝑁}) ∖ {𝑀}) |
13 | 8, 12 | syl6sseqr 3615 | 1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑀 ∉ (〈𝑉, 𝐸〉 Neighbors 𝑁)) → (〈𝑉, 𝐸〉 Neighbors 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 {cpr 4127 〈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: nbhashuvtx1 26442 |
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