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Theorem iscusgr 27127
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
StepHypRef Expression
1 df-cusgr 27121 . 2 ComplUSGraph = (USGraph ∩ ComplGraph)
21elin2 4171 1 (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  USGraphcusgr 26861  ComplGraphccplgr 27118  ComplUSGraphccusgr 27119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-cusgr 27121
This theorem is referenced by:  cusgrusgr  27128  cusgrcplgr  27129  iscusgrvtx  27130  cusgruvtxb  27131  iscusgredg  27132  cusgr0  27135  cusgr0v  27137  cusgr1v  27140  cusgrop  27147  cusgrexi  27152  structtocusgr  27155  cusgrres  27157
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