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Mirrors > Home > MPE Home > Th. List > nbgrcl | Structured version Visualization version GIF version |
Description: If a class has at least one neighbor, it must be a vertex. (Contributed by AV, 6-Jun-2021.) |
Ref | Expression |
---|---|
nbgrcl | ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nbgr 26228 | . . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) | |
2 | 1 | mpt2xeldm 7337 | . 2 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))) |
3 | csbfv 6233 | . . . 4 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺) | |
4 | 3 | eleq2i 2693 | . . 3 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) ↔ 𝑋 ∈ (Vtx‘𝐺)) |
5 | 4 | biimpi 206 | . 2 ⊢ (𝑋 ∈ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → 𝑋 ∈ (Vtx‘𝐺)) |
6 | 2, 5 | simpl2im 658 | 1 ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∃wrex 2913 {crab 2916 Vcvv 3200 ⦋csb 3533 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 {cpr 4179 ‘cfv 5888 (class class class)co 6650 Vtxcvtx 25874 Edgcedg 25939 NeighbVtx cnbgr 26224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-nbgr 26228 |
This theorem is referenced by: nbgrel 26238 frgrnbnb 27157 |
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