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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 2906 | . . . 4 ⊢ (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ (Vtx‘𝐺)) |
3 | fveq2 6672 | . . . . . 6 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = (Vtx‘∅)) | |
4 | vtxval0 26826 | . . . . . 6 ⊢ (Vtx‘∅) = ∅ | |
5 | 3, 4 | syl6eq 2874 | . . . . 5 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = ∅) |
6 | 5 | eleq2d 2900 | . . . 4 ⊢ (𝐺 = ∅ → (𝑈 ∈ (Vtx‘𝐺) ↔ 𝑈 ∈ ∅)) |
7 | 2, 6 | syl5bb 285 | . . 3 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ ∅)) |
8 | noel 4298 | . . . 4 ⊢ ¬ 𝑈 ∈ ∅ | |
9 | 8 | pm2.21i 119 | . . 3 ⊢ (𝑈 ∈ ∅ → ((VtxDeg‘𝐺)‘𝑈) = 0) |
10 | 7, 9 | syl6bi 255 | . 2 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0)) |
11 | 10 | imp 409 | 1 ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∅c0 4293 ‘cfv 6357 0cc0 10539 Vtxcvtx 26783 VtxDegcvtxdg 27249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-slot 16489 df-base 16491 df-vtx 26785 |
This theorem is referenced by: (None) |
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