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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 6671 | . . 3 ⊢ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} | |
2 | dm0 5774 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6589 | . . . 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 234 | . 2 ⊢ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} |
6 | 0ex 5185 | . . 3 ⊢ ∅ ∈ V | |
7 | vtxval0 27084 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
8 | 7 | eqcomi 2745 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
9 | iedgval0 27085 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
10 | 9 | eqcomi 2745 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
11 | 8, 10 | isusgr 27198 | . . 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 234 | 1 ⊢ ∅ ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2112 {crab 3055 Vcvv 3398 ∖ cdif 3850 ∅c0 4223 𝒫 cpw 4499 {csn 4527 dom cdm 5536 –1-1→wf1 6355 ‘cfv 6358 2c2 11850 ♯chash 13861 Vtxcvtx 27041 iEdgciedg 27042 USGraphcusgr 27194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-nn 11796 df-ndx 16669 df-slot 16670 df-base 16672 df-edgf 27034 df-vtx 27043 df-iedg 27044 df-usgr 27196 |
This theorem is referenced by: cusgr0 27468 frgr0 28302 |
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