Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vtxdg0v | Structured version Visualization version GIF version |
Description: The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.) |
Ref | Expression |
---|---|
vtxdgf.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
vtxdg0v | ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdgf.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | eleq2i 2881 | . . . 4 ⊢ (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ (Vtx‘𝐺)) |
3 | fveq2 6645 | . . . . . 6 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = (Vtx‘∅)) | |
4 | vtxval0 26832 | . . . . . 6 ⊢ (Vtx‘∅) = ∅ | |
5 | 3, 4 | eqtrdi 2849 | . . . . 5 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = ∅) |
6 | 5 | eleq2d 2875 | . . . 4 ⊢ (𝐺 = ∅ → (𝑈 ∈ (Vtx‘𝐺) ↔ 𝑈 ∈ ∅)) |
7 | 2, 6 | syl5bb 286 | . . 3 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ ∅)) |
8 | noel 4247 | . . . 4 ⊢ ¬ 𝑈 ∈ ∅ | |
9 | 8 | pm2.21i 119 | . . 3 ⊢ (𝑈 ∈ ∅ → ((VtxDeg‘𝐺)‘𝑈) = 0) |
10 | 7, 9 | syl6bi 256 | . 2 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0)) |
11 | 10 | imp 410 | 1 ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∅c0 4243 ‘cfv 6324 0cc0 10526 Vtxcvtx 26789 VtxDegcvtxdg 27255 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-slot 16479 df-base 16481 df-vtx 26791 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |