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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 6640 | . . 3 ⊢ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} | |
2 | dm0 5783 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6564 | . . . 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 233 | . 2 ⊢ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} |
6 | 0ex 5202 | . . 3 ⊢ ∅ ∈ V | |
7 | vtxval0 26816 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
8 | 7 | eqcomi 2828 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
9 | iedgval0 26817 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
10 | 9 | eqcomi 2828 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
11 | 8, 10 | isusgr 26930 | . . 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 233 | 1 ⊢ ∅ ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1531 ∈ wcel 2108 {crab 3140 Vcvv 3493 ∖ cdif 3931 ∅c0 4289 𝒫 cpw 4537 {csn 4559 dom cdm 5548 –1-1→wf1 6345 ‘cfv 6348 2c2 11684 ♯chash 13682 Vtxcvtx 26773 iEdgciedg 26774 USGraphcusgr 26926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fv 6356 df-slot 16479 df-base 16481 df-edgf 26767 df-vtx 26775 df-iedg 26776 df-usgr 26928 |
This theorem is referenced by: cusgr0 27200 frgr0 28036 |
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