Theorem cusgra0v 25989

Description: A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)

Assertion

Ref	Expression
cusgra0v	⊢ ∅ ComplUSGrph ∅

Proof of Theorem cusgra0v

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4718	. 2 ⊢ ∅ ∈ V
2		usgra0 25899	. . . 4 ⊢ (∅ ∈ V → ∅ USGrph ∅)
3	1, 2	mp1i 13	. . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → ∅ USGrph ∅)
4		ral0 4028	. . . 4 ⊢ ∀𝑘 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅
5	4	a1i 11	. . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → ∀𝑘 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅)
6		iscusgra 25985	. . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ComplUSGrph ∅ ↔ (∅ USGrph ∅ ∧ ∀𝑘 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅)))
7	3, 5, 6	mpbir2and 959	. 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → ∅ ComplUSGrph ∅)
8	1, 1, 7	mp2an 704	1 ⊢ ∅ ComplUSGrph ∅

Syntax hints:   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   ComplUSGrph ccusgra 25947

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-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-cusgra 25950

This theorem is referenced by:  cusgra1v  25990