Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nbgrnself2 | Structured version Visualization version GIF version |
Description: A class 𝑋 is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.) |
Ref | Expression |
---|---|
nbgrnself2 | ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑣 = 𝑋 → 𝑣 = 𝑋) | |
2 | oveq2 7163 | . . . 4 ⊢ (𝑣 = 𝑋 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑋)) | |
3 | 1, 2 | neleq12d 3127 | . . 3 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋))) |
4 | eqid 2821 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | 4 | nbgrnself 27140 | . . 3 ⊢ ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣) |
6 | 3, 5 | vtoclri 3584 | . 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋)) |
7 | 4 | nbgrisvtx 27122 | . . . 4 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺)) |
8 | 7 | con3i 157 | . . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋)) |
9 | df-nel 3124 | . . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋)) | |
10 | 8, 9 | sylibr 236 | . 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋)) |
11 | 6, 10 | pm2.61i 184 | 1 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 ∉ wnel 3123 ‘cfv 6354 (class class class)co 7155 Vtxcvtx 26780 NeighbVtx cnbgr 27113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-nbgr 27114 |
This theorem is referenced by: nbgrssovtx 27142 nb3grprlem2 27162 |
Copyright terms: Public domain | W3C validator |