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Theorem usgr0 27033
 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 6622 . . 3 ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}
2 dm0 5754 . . . 4 dom ∅ = ∅
3 f1eq2 6545 . . . 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 234 . 2 ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}
6 0ex 5175 . . 3 ∅ ∈ V
7 vtxval0 26832 . . . . 5 (Vtx‘∅) = ∅
87eqcomi 2807 . . . 4 ∅ = (Vtx‘∅)
9 iedgval0 26833 . . . . 5 (iEdg‘∅) = ∅
109eqcomi 2807 . . . 4 ∅ = (iEdg‘∅)
118, 10isusgr 26946 . . 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 234 1 ∅ ∈ USGraph
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  𝒫 cpw 4497  {csn 4525  dom cdm 5519  –1-1→wf1 6321  ‘cfv 6324  2c2 11680  ♯chash 13686  Vtxcvtx 26789  iEdgciedg 26790  USGraphcusgr 26942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fv 6332  df-slot 16479  df-base 16481  df-edgf 26783  df-vtx 26791  df-iedg 26792  df-usgr 26944 This theorem is referenced by:  cusgr0  27216  frgr0  28050
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