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Theorem usgr0 26357
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 6310 . . 3 ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}
2 dm0 5477 . . . 4 dom ∅ = ∅
3 f1eq2 6237 . . . 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 221 . 2 ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}
6 0ex 4921 . . 3 ∅ ∈ V
7 vtxval0 26151 . . . . 5 (Vtx‘∅) = ∅
87eqcomi 2779 . . . 4 ∅ = (Vtx‘∅)
9 iedgval0 26152 . . . . 5 (iEdg‘∅) = ∅
109eqcomi 2779 . . . 4 ∅ = (iEdg‘∅)
118, 10isusgr 26269 . . 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 221 1 ∅ ∈ USGraph
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1630  wcel 2144  {crab 3064  Vcvv 3349  cdif 3718  c0 4061  𝒫 cpw 4295  {csn 4314  dom cdm 5249  1-1wf1 6028  cfv 6031  2c2 11271  chash 13320  Vtxcvtx 26094  iEdgciedg 26095  USGraphcusgr 26265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fv 6039  df-slot 16067  df-base 16069  df-edgf 26088  df-vtx 26096  df-iedg 26097  df-usgr 26267
This theorem is referenced by:  cusgr0  26556  frgr0  27443
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