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Theorem usgr0 26744
Description: The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
Assertion
Ref Expression
usgr0 ∅ ∈ USGraph

Proof of Theorem usgr0
StepHypRef 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}))
42, 3ax-mp 5 . . 3 (∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2} ↔ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2})
51, 4mpbir 223 . 2 ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}
6 0ex 5065 . . 3 ∅ ∈ V
7 vtxval0 26543 . . . . 5 (Vtx‘∅) = ∅
87eqcomi 2782 . . . 4 ∅ = (Vtx‘∅)
9 iedgval0 26544 . . . . 5 (iEdg‘∅) = ∅
109eqcomi 2782 . . . 4 ∅ = (iEdg‘∅)
118, 10isusgr 26657 . . 3 (∅ ∈ V → (∅ ∈ USGraph ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}))
126, 11ax-mp 5 . 2 (∅ ∈ USGraph ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2})
135, 12mpbir 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  1-1wf1 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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