Theorem frgr0 28047

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 27028	. 2 ⊢ ∅ ∈ USGraph
2		ral0 4459	. 2 ⊢ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)
3		vtxval0 26827	. . . 4 ⊢ (Vtx‘∅) = ∅
4	3	eqcomi 2833	. . 3 ⊢ ∅ = (Vtx‘∅)
5		eqid 2824	. . 3 ⊢ (Edg‘∅) = (Edg‘∅)
6	4, 5	isfrgr 28042	. 2 ⊢ (∅ ∈ FriendGraph ↔ (∅ ∈ USGraph ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅)))
7	1, 2, 6	mpbir2an 709	1 ⊢ ∅ ∈ FriendGraph

Syntax hints:   ∈ wcel 2113  ∀wral 3141  ∃!wreu 3143   ∖ cdif 3936   ⊆ wss 3939  ∅c0 4294  {csn 4570  {cpr 4572  ‘cfv 6358  Vtxcvtx 26784  Edgcedg 26835  USGraphcusgr 26937   FriendGraph cfrgr 28040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fv 6366  df-slot 16490  df-base 16492  df-edgf 26778  df-vtx 26786  df-iedg 26787  df-usgr 26939  df-frgr 28041

This theorem is referenced by: (None)