Theorem usgr0 27025

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

Step	Hyp	Ref	Expression
1		f10 6647	. . 3 ⊢ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}
2		dm0 5790	. . . 4 ⊢ dom ∅ = ∅
3		f1eq2 6571	. . . 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 233	. 2 ⊢ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (♯‘𝑥) = 2}
6		0ex 5211	. . 3 ⊢ ∅ ∈ V
7		vtxval0 26824	. . . . 5 ⊢ (Vtx‘∅) = ∅
8	7	eqcomi 2830	. . . 4 ⊢ ∅ = (Vtx‘∅)
9		iedgval0 26825	. . . . 5 ⊢ (iEdg‘∅) = ∅
10	9	eqcomi 2830	. . . 4 ⊢ ∅ = (iEdg‘∅)
11	8, 10	isusgr 26938	. . 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 233	1 ⊢ ∅ ∈ USGraph

Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114  {crab 3142  Vcvv 3494   ∖ cdif 3933  ∅c0 4291  𝒫 cpw 4539  {csn 4567  dom cdm 5555  –1-1→wf1 6352  ‘cfv 6355  2c2 11693  ♯chash 13691  Vtxcvtx 26781  iEdgciedg 26782  USGraphcusgr 26934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fv 6363  df-slot 16487  df-base 16489  df-edgf 26775  df-vtx 26783  df-iedg 26784  df-usgr 26936

This theorem is referenced by:  cusgr0  27208  frgr0  28044