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Theorem iscusgr 40640
 Description: The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
Assertion
Ref Expression
iscusgr (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Proof of Theorem iscusgr
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2676 . 2 (𝑔 = 𝐺 → (𝑔 ∈ ComplGraph ↔ 𝐺 ∈ ComplGraph))
2 df-cusgr 40558 . 2 ComplUSGraph = {𝑔 ∈ USGraph ∣ 𝑔 ∈ ComplGraph}
31, 2elrab2 3333 1 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   USGraph cusgr 40379  ComplGraphccplgr 40552  ComplUSGraphccusgr 40553 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-cusgr 40558 This theorem is referenced by:  cusgrusgr  40641  cusgrcplgr  40642  iscusgrvtx  40643  cusgruvtxb  40644  iscusgredg  40645  cusgr0  40648  cusgr0v  40650  cusgr1v  40653  cusgrop  40660  cusgrexi  40662  cusgrres  40664
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