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Mirrors > Home > MPE Home > Th. List > nbgr0vtxlem | Structured version Visualization version GIF version |
Description: Lemma for nbgr0vtx 26841 and nbgr0edg 26842. (Contributed by AV, 15-Nov-2020.) |
Ref | Expression |
---|---|
nbgr0vtxlem.v | ⊢ (𝜑 → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒) |
Ref | Expression |
---|---|
nbgr0vtxlem | ⊢ (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2778 | . . . . . . . 8 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | nbgrval 26821 | . . . . . . 7 ⊢ (𝐾 ∈ (Vtx‘𝐺) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒}) |
4 | 3 | ad2antrl 715 | . . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝐾 ∈ V) ∧ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝜑)) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒}) |
5 | nbgr0vtxlem.v | . . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒) | |
6 | 5 | ad2antll 716 | . . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝐾 ∈ V) ∧ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝜑)) → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒) |
7 | rabeq0 4224 | . . . . . . 7 ⊢ ({𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒) | |
8 | 6, 7 | sylibr 226 | . . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝐾 ∈ V) ∧ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝜑)) → {𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒} = ∅) |
9 | 4, 8 | eqtrd 2814 | . . . . 5 ⊢ (((𝐺 ∈ V ∧ 𝐾 ∈ V) ∧ (𝐾 ∈ (Vtx‘𝐺) ∧ 𝜑)) → (𝐺 NeighbVtx 𝐾) = ∅) |
10 | 9 | expcom 406 | . . . 4 ⊢ ((𝐾 ∈ (Vtx‘𝐺) ∧ 𝜑) → ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 NeighbVtx 𝐾) = ∅)) |
11 | 10 | ex 405 | . . 3 ⊢ (𝐾 ∈ (Vtx‘𝐺) → (𝜑 → ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 NeighbVtx 𝐾) = ∅))) |
12 | 11 | com23 86 | . 2 ⊢ (𝐾 ∈ (Vtx‘𝐺) → ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅))) |
13 | df-nel 3074 | . . . 4 ⊢ (𝐾 ∉ (Vtx‘𝐺) ↔ ¬ 𝐾 ∈ (Vtx‘𝐺)) | |
14 | 1 | nbgrnvtx0 26824 | . . . 4 ⊢ (𝐾 ∉ (Vtx‘𝐺) → (𝐺 NeighbVtx 𝐾) = ∅) |
15 | 13, 14 | sylbir 227 | . . 3 ⊢ (¬ 𝐾 ∈ (Vtx‘𝐺) → (𝐺 NeighbVtx 𝐾) = ∅) |
16 | 15 | a1d 25 | . 2 ⊢ (¬ 𝐾 ∈ (Vtx‘𝐺) → (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅)) |
17 | nbgrprc0 26819 | . . 3 ⊢ (¬ (𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 NeighbVtx 𝐾) = ∅) | |
18 | 17 | a1d 25 | . 2 ⊢ (¬ (𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅)) |
19 | 12, 16, 18 | pm2.61nii 179 | 1 ⊢ (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∉ wnel 3073 ∀wral 3088 ∃wrex 3089 {crab 3092 Vcvv 3415 ∖ cdif 3826 ⊆ wss 3829 ∅c0 4178 {csn 4441 {cpr 4443 ‘cfv 6188 (class class class)co 6976 Vtxcvtx 26484 Edgcedg 26535 NeighbVtx cnbgr 26817 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
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 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-nel 3074 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-1st 7501 df-2nd 7502 df-nbgr 26818 |
This theorem is referenced by: nbgr0vtx 26841 nbgr0edg 26842 nbgr1vtx 26843 |
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