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Mirrors > Home > MPE Home > Th. List > nbgr0vtx | Structured version Visualization version GIF version |
Description: In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) |
Ref | Expression |
---|---|
nbgr0vtx | ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4270 | . . 3 ⊢ ∀𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒 | |
2 | difeq1 3920 | . . . . 5 ⊢ ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = (∅ ∖ {𝐾})) | |
3 | 0dif 4174 | . . . . 5 ⊢ (∅ ∖ {𝐾}) = ∅ | |
4 | 2, 3 | syl6eq 2850 | . . . 4 ⊢ ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = ∅) |
5 | 4 | raleqdv 3328 | . . 3 ⊢ ((Vtx‘𝐺) = ∅ → (∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒 ↔ ∀𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)) |
6 | 1, 5 | mpbiri 250 | . 2 ⊢ ((Vtx‘𝐺) = ∅ → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒) |
7 | 6 | nbgr0vtxlem 26592 | 1 ⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1653 ∀wral 3090 ∃wrex 3091 ∖ cdif 3767 ⊆ wss 3770 ∅c0 4116 {csn 4369 {cpr 4371 ‘cfv 6102 (class class class)co 6879 Vtxcvtx 26230 Edgcedg 26281 NeighbVtx cnbgr 26565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-1st 7402 df-2nd 7403 df-nbgr 26566 |
This theorem is referenced by: (None) |
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