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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 6866 | . . 3 ⊢ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} | |
2 | dm0 5917 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6783 | . . . 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 230 | . 2 ⊢ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} |
6 | 0ex 5301 | . . 3 ⊢ ∅ ∈ V | |
7 | vtxval0 28845 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
8 | 7 | eqcomi 2737 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
9 | iedgval0 28846 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
10 | 9 | eqcomi 2737 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
11 | 8, 10 | isusgr 28959 | . . 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 230 | 1 ⊢ ∅ ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 {crab 3428 Vcvv 3470 ∖ cdif 3942 ∅c0 4318 𝒫 cpw 4598 {csn 4624 dom cdm 5672 –1-1→wf1 6539 ‘cfv 6542 2c2 12291 ♯chash 14315 Vtxcvtx 28802 iEdgciedg 28803 USGraphcusgr 28955 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-dec 12702 df-slot 17144 df-ndx 17156 df-base 17174 df-edgf 28793 df-vtx 28804 df-iedg 28805 df-usgr 28957 |
This theorem is referenced by: cusgr0 29232 frgr0 30068 |
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