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Mirrors > Home > MPE Home > Th. List > usgr0 | Structured version Visualization version GIF version |
Description: The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.) |
Ref | Expression |
---|---|
usgr0 | ⊢ ∅ ∈ USGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f10 6474 | . . 3 ⊢ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} | |
2 | dm0 5635 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6398 | . . . 4 ⊢ (dom ∅ = ∅ → (∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2})) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
5 | 1, 4 | mpbir 223 | . 2 ⊢ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} |
6 | 0ex 5065 | . . 3 ⊢ ∅ ∈ V | |
7 | vtxval0 26543 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
8 | 7 | eqcomi 2782 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
9 | iedgval0 26544 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
10 | 9 | eqcomi 2782 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
11 | 8, 10 | isusgr 26657 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ USGraph ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
12 | 6, 11 | ax-mp 5 | . 2 ⊢ (∅ ∈ USGraph ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
13 | 5, 12 | mpbir 223 | 1 ⊢ ∅ ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1508 ∈ wcel 2051 {crab 3087 Vcvv 3410 ∖ cdif 3821 ∅c0 4173 𝒫 cpw 4417 {csn 4436 dom cdm 5404 –1-1→wf1 6183 ‘cfv 6186 2c2 11494 ♯chash 13504 Vtxcvtx 26500 iEdgciedg 26501 USGraphcusgr 26653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fv 6194 df-slot 16342 df-base 16344 df-edgf 26494 df-vtx 26502 df-iedg 26503 df-usgr 26655 |
This theorem is referenced by: cusgr0 26927 frgr0 27814 |
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