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Theorem cusgra1v 25990
 Description: A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
Assertion
Ref Expression
cusgra1v {𝐴} ComplUSGrph ∅

Proof of Theorem cusgra1v
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4835 . . . 4 {𝐴} ∈ V
2 usgra0 25899 . . . 4 ({𝐴} ∈ V → {𝐴} USGrph ∅)
31, 2mp1i 13 . . 3 (𝐴 ∈ V → {𝐴} USGrph ∅)
4 ral0 4028 . . . 4 𝑛 ∈ ∅ {𝑛, 𝐴} ∈ ran ∅
5 sneq 4135 . . . . . . . 8 (𝑘 = 𝐴 → {𝑘} = {𝐴})
65difeq2d 3690 . . . . . . 7 (𝑘 = 𝐴 → ({𝐴} ∖ {𝑘}) = ({𝐴} ∖ {𝐴}))
7 difid 3902 . . . . . . 7 ({𝐴} ∖ {𝐴}) = ∅
86, 7syl6eq 2660 . . . . . 6 (𝑘 = 𝐴 → ({𝐴} ∖ {𝑘}) = ∅)
9 preq2 4213 . . . . . . 7 (𝑘 = 𝐴 → {𝑛, 𝑘} = {𝑛, 𝐴})
109eleq1d 2672 . . . . . 6 (𝑘 = 𝐴 → ({𝑛, 𝑘} ∈ ran ∅ ↔ {𝑛, 𝐴} ∈ ran ∅))
118, 10raleqbidv 3129 . . . . 5 (𝑘 = 𝐴 → (∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅ ↔ ∀𝑛 ∈ ∅ {𝑛, 𝐴} ∈ ran ∅))
1211ralsng 4165 . . . 4 (𝐴 ∈ V → (∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅ ↔ ∀𝑛 ∈ ∅ {𝑛, 𝐴} ∈ ran ∅))
134, 12mpbiri 247 . . 3 (𝐴 ∈ V → ∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅)
14 0ex 4718 . . . 4 ∅ ∈ V
15 iscusgra 25985 . . . 4 (({𝐴} ∈ V ∧ ∅ ∈ V) → ({𝐴} ComplUSGrph ∅ ↔ ({𝐴} USGrph ∅ ∧ ∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅)))
161, 14, 15mp2an 704 . . 3 ({𝐴} ComplUSGrph ∅ ↔ ({𝐴} USGrph ∅ ∧ ∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅))
173, 13, 16sylanbrc 695 . 2 (𝐴 ∈ V → {𝐴} ComplUSGrph ∅)
18 snprc 4197 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
19 cusgra0v 25989 . . . 4 ∅ ComplUSGrph ∅
20 breq1 4586 . . . 4 ({𝐴} = ∅ → ({𝐴} ComplUSGrph ∅ ↔ ∅ ComplUSGrph ∅))
2119, 20mpbiri 247 . . 3 ({𝐴} = ∅ → {𝐴} ComplUSGrph ∅)
2218, 21sylbi 206 . 2 𝐴 ∈ V → {𝐴} ComplUSGrph ∅)
2317, 22pm2.61i 175 1 {𝐴} ComplUSGrph ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   ComplUSGrph ccusgra 25947 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-usgra 25862  df-cusgra 25950 This theorem is referenced by: (None)
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