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Theorem usgr0 26062
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 6136 . . 3 ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2}
2 dm0 5309 . . . 4 dom ∅ = ∅
3 f1eq2 6064 . . . 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 4760 . . 3 ∅ ∈ V
7 vtxval0 25865 . . . . 5 (Vtx‘∅) = ∅
87eqcomi 2630 . . . 4 ∅ = (Vtx‘∅)
9 iedgval0 25866 . . . . 5 (iEdg‘∅) = ∅
109eqcomi 2630 . . . 4 ∅ = (iEdg‘∅)
118, 10isusgr 25975 . . 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 1480  wcel 1987  {crab 2912  Vcvv 3190  cdif 3557  c0 3897  𝒫 cpw 4136  {csn 4155  dom cdm 5084  1-1wf1 5854  cfv 5857  2c2 11030  #chash 13073  Vtxcvtx 25808  iEdgciedg 25809   USGraph cusgr 25971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fv 5865  df-slot 15804  df-base 15805  df-edgf 25802  df-vtx 25810  df-iedg 25811  df-usgr 25973
This theorem is referenced by:  cusgr0  26243  frgr0  27028
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