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Theorem frgra0v 26520
 Description: Any graph with no vertex is a friendship graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra0v (∅ FriendGrph 𝐸𝐸 = ∅)

Proof of Theorem frgra0v
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 26519 . . 3 (∅ FriendGrph 𝐸 → ∅ USGrph 𝐸)
2 usgra0v 25900 . . 3 (∅ USGrph 𝐸𝐸 = ∅)
31, 2sylib 207 . 2 (∅ FriendGrph 𝐸𝐸 = ∅)
42biimpri 217 . . 3 (𝐸 = ∅ → ∅ USGrph 𝐸)
5 ral0 4028 . . . 4 𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸
65a1i 11 . . 3 (𝐸 = ∅ → ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
7 0ex 4718 . . . 4 ∅ ∈ V
8 eleq1 2676 . . . . 5 (𝐸 = ∅ → (𝐸 ∈ V ↔ ∅ ∈ V))
97, 8mpbiri 247 . . . 4 (𝐸 = ∅ → 𝐸 ∈ V)
10 isfrgra 26517 . . . 4 ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ FriendGrph 𝐸 ↔ (∅ USGrph 𝐸 ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
117, 9, 10sylancr 694 . . 3 (𝐸 = ∅ → (∅ FriendGrph 𝐸 ↔ (∅ USGrph 𝐸 ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
124, 6, 11mpbir2and 959 . 2 (𝐸 = ∅ → ∅ FriendGrph 𝐸)
133, 12impbii 198 1 (∅ FriendGrph 𝐸𝐸 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   FriendGrph cfrgra 26515 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-reu 2903  df-rab 2905  df-v 3175  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-frgra 26516 This theorem is referenced by:  frgra0  26521
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