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Theorem frgr0 27446
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 26358 . 2 ∅ ∈ USGraph
2 ral0 4217 . 2 𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)
3 vtxval0 26152 . . . 4 (Vtx‘∅) = ∅
43eqcomi 2780 . . 3 ∅ = (Vtx‘∅)
5 eqid 2771 . . 3 (Edg‘∅) = (Edg‘∅)
64, 5frgrusgrfrcond 27441 . 2 (∅ ∈ FriendGraph ↔ (∅ ∈ USGraph ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)))
71, 2, 6mpbir2an 690 1 ∅ ∈ FriendGraph
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  wral 3061  ∃!wreu 3063  cdif 3720  wss 3723  c0 4063  {csn 4316  {cpr 4318  cfv 6031  Vtxcvtx 26095  Edgcedg 26160  USGraphcusgr 26266   FriendGraph cfrgr 27438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fv 6039  df-slot 16068  df-base 16070  df-edgf 26089  df-vtx 26097  df-iedg 26098  df-usgr 26268  df-frgr 27439
This theorem is referenced by: (None)
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