Theorem frgr0 28028

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.

Step	Hyp	Ref	Expression
1		usgr0 27011	. 2 ⊢ ∅ ∈ USGraph
2		ral0 4442	. 2 ⊢ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)
3		vtxval0 26810	. . . 4 ⊢ (Vtx‘∅) = ∅
4	3	eqcomi 2830	. . 3 ⊢ ∅ = (Vtx‘∅)
5		eqid 2821	. . 3 ⊢ (Edg‘∅) = (Edg‘∅)
6	4, 5	isfrgr 28023	. 2 ⊢ (∅ ∈ FriendGraph ↔ (∅ ∈ USGraph ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)))
7	1, 2, 6	mpbir2an 709	1 ⊢ ∅ ∈ FriendGraph

Syntax hints:   ∈ wcel 2114  ∀wral 3138  ∃!wreu 3140   ∖ cdif 3921   ⊆ wss 3924  ∅c0 4279  {csn 4553  {cpr 4555  ‘cfv 6341  Vtxcvtx 26767  Edgcedg 26818  USGraphcusgr 26920   FriendGraph cfrgr 28021

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 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316

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-reu 3145  df-rab 3147  df-v 3488  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fv 6349  df-slot 16470  df-base 16472  df-edgf 26761  df-vtx 26769  df-iedg 26770  df-usgr 26922  df-frgr 28022

This theorem is referenced by: (None)