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Theorem frgr0 28053
Description: The null graph (graph without vertices) is a friendship graph. (Contributed by AV, 29-Mar-2021.)
Assertion
Ref Expression
frgr0 ∅ ∈ FriendGraph

Proof of Theorem frgr0
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgr0 27036 . 2 ∅ ∈ USGraph
2 ral0 4439 . 2 𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)
3 vtxval0 26835 . . . 4 (Vtx‘∅) = ∅
43eqcomi 2833 . . 3 ∅ = (Vtx‘∅)
5 eqid 2824 . . 3 (Edg‘∅) = (Edg‘∅)
64, 5isfrgr 28048 . 2 (∅ ∈ FriendGraph ↔ (∅ ∈ USGraph ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)))
71, 2, 6mpbir2an 710 1 ∅ ∈ FriendGraph
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  wral 3133  ∃!wreu 3135  cdif 3916  wss 3919  c0 4276  {csn 4550  {cpr 4552  cfv 6343  Vtxcvtx 26792  Edgcedg 26843  USGraphcusgr 26945   FriendGraph cfrgr 28046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fv 6351  df-slot 16487  df-base 16489  df-edgf 26786  df-vtx 26794  df-iedg 26795  df-usgr 26947  df-frgr 28047
This theorem is referenced by: (None)
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