![]() |
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 7401 | . . . 4 ⊢ (𝑣 = 𝑋 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑋)) | |
3 | 1, 2 | neleq12d 3050 | . . 3 ⊢ (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋))) |
4 | eqid 2731 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | 4 | nbgrnself 28481 | . . 3 ⊢ ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣) |
6 | 3, 5 | vtoclri 3573 | . 2 ⊢ (𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋)) |
7 | 4 | nbgrisvtx 28463 | . . . 4 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺)) |
8 | 7 | con3i 154 | . . 3 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋)) |
9 | df-nel 3046 | . . 3 ⊢ (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋)) | |
10 | 8, 9 | sylibr 233 | . 2 ⊢ (¬ 𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋)) |
11 | 6, 10 | pm2.61i 182 | 1 ⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 ∉ wnel 3045 ‘cfv 6532 (class class class)co 7393 Vtxcvtx 28121 NeighbVtx cnbgr 28454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-nbgr 28455 |
This theorem is referenced by: nbgrssovtx 28483 nb3grprlem2 28503 |
Copyright terms: Public domain | W3C validator |