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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 2680 | . . . 4 ⊢ (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ (Vtx‘𝐺)) |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = (Vtx‘∅)) | |
4 | vtxval0 25714 | . . . . . 6 ⊢ (Vtx‘∅) = ∅ | |
5 | 3, 4 | syl6eq 2660 | . . . . 5 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = ∅) |
6 | 5 | eleq2d 2673 | . . . 4 ⊢ (𝐺 = ∅ → (𝑈 ∈ (Vtx‘𝐺) ↔ 𝑈 ∈ ∅)) |
7 | 2, 6 | syl5bb 271 | . . 3 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ ∅)) |
8 | noel 3878 | . . . 4 ⊢ ¬ 𝑈 ∈ ∅ | |
9 | 8 | pm2.21i 115 | . . 3 ⊢ (𝑈 ∈ ∅ → ((VtxDeg‘𝐺)‘𝑈) = 0) |
10 | 7, 9 | syl6bi 242 | . 2 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0)) |
11 | 10 | imp 444 | 1 ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ‘cfv 5804 0cc0 9815 Vtxcvtx 25673 VtxDegcvtxdg 40681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-slot 15699 df-base 15700 df-vtx 25675 |
This theorem is referenced by: (None) |
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