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Mirrors > Home > MPE Home > Th. List > nbgrnvtx0 | Structured version Visualization version GIF version |
Description: If a class 𝑋 is not a vertex of a graph 𝐺, then it has no neighbors in 𝐺. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
Ref | Expression |
---|---|
nbgrel.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
nbgrnvtx0 | ⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgrel.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | csbfv 6539 | . . . . . 6 ⊢ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) = (Vtx‘𝐺) | |
3 | 1, 2 | eqtr4i 2799 | . . . . 5 ⊢ 𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) |
4 | neleq2 3073 | . . . . 5 ⊢ (𝑉 = ⦋𝐺 / 𝑔⦌(Vtx‘𝑔) → (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) |
6 | 5 | biimpi 208 | . . 3 ⊢ (𝑋 ∉ 𝑉 → 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) |
7 | 6 | olcd 860 | . 2 ⊢ (𝑋 ∉ 𝑉 → (𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔))) |
8 | df-nbgr 26808 | . . 3 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) | |
9 | 8 | mpoxneldm 7674 | . 2 ⊢ ((𝐺 ∉ V ∨ 𝑋 ∉ ⦋𝐺 / 𝑔⦌(Vtx‘𝑔)) → (𝐺 NeighbVtx 𝑋) = ∅) |
10 | 7, 9 | syl 17 | 1 ⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∨ wo 833 = wceq 1507 ∉ wnel 3067 ∃wrex 3083 {crab 3086 Vcvv 3409 ⦋csb 3782 ∖ cdif 3822 ⊆ wss 3825 ∅c0 4173 {csn 4435 {cpr 4437 ‘cfv 6182 (class class class)co 6970 Vtxcvtx 26474 Edgcedg 26525 NeighbVtx cnbgr 26807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-nel 3068 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-nbgr 26808 |
This theorem is referenced by: nbuhgr 26818 nbumgr 26822 nbgr0vtxlem 26830 nbgr1vtx 26833 |
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