Theorem cusisusgra 25987

Description: A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)

Assertion

Ref	Expression
cusisusgra	⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 USGrph 𝐸)

Proof of Theorem cusisusgra

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

Step	Hyp	Ref	Expression
1		iscusgra0 25986	. 2 ⊢ (𝑉 ComplUSGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
2	1	simpld 474	1 ⊢ (𝑉 ComplUSGrph 𝐸 → 𝑉 USGrph 𝐸)

Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  {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-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-cusgra 25950

This theorem is referenced by:  nbcusgra  25992  cusgrasizeindb0  25999  cusgrasizeindb1  26000  cusgrares  26001  cusgrasizeindslem2  26003  cusgrasizeinds  26004  cusgrasize2inds  26005  cusgrafi  26010  sizeusglecusg  26014  cusconngra  26204  cusgraisrusgra  26465