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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 2693 | . . . 4 ⊢ (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ (Vtx‘𝐺)) |
3 | fveq2 6191 | . . . . . 6 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = (Vtx‘∅)) | |
4 | vtxval0 25931 | . . . . . 6 ⊢ (Vtx‘∅) = ∅ | |
5 | 3, 4 | syl6eq 2672 | . . . . 5 ⊢ (𝐺 = ∅ → (Vtx‘𝐺) = ∅) |
6 | 5 | eleq2d 2687 | . . . 4 ⊢ (𝐺 = ∅ → (𝑈 ∈ (Vtx‘𝐺) ↔ 𝑈 ∈ ∅)) |
7 | 2, 6 | syl5bb 272 | . . 3 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 ↔ 𝑈 ∈ ∅)) |
8 | noel 3919 | . . . 4 ⊢ ¬ 𝑈 ∈ ∅ | |
9 | 8 | pm2.21i 116 | . . 3 ⊢ (𝑈 ∈ ∅ → ((VtxDeg‘𝐺)‘𝑈) = 0) |
10 | 7, 9 | syl6bi 243 | . 2 ⊢ (𝐺 = ∅ → (𝑈 ∈ 𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0)) |
11 | 10 | imp 445 | 1 ⊢ ((𝐺 = ∅ ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 ‘cfv 5888 0cc0 9936 Vtxcvtx 25874 VtxDegcvtxdg 26361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-slot 15861 df-base 15863 df-vtx 25876 |
This theorem is referenced by: (None) |
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