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Theorem isausgra 24017
Description: The property of an unordered pair to be an alternatively defined undirected simple graph without loops (defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
Hypothesis
Ref Expression
ausgra.1  |-  G  =  { <. v ,  e
>.  |  e  C_  { x  e.  ~P v  |  ( # `  x
)  =  2 } }
Assertion
Ref Expression
isausgra  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V G E  <-> 
E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } ) )
Distinct variable groups:    v, e, x, E    e, V, v, x    x, X
Allowed substitution hints:    G( x, v, e)    W( x, v, e)    X( v, e)

Proof of Theorem isausgra
StepHypRef Expression
1 simpr 461 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  e  =  E )
2 pweq 4006 . . . . 5  |-  ( v  =  V  ->  ~P v  =  ~P V
)
32adantr 465 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
4 biidd 237 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( # `  x
)  =  2  <->  ( # `
 x )  =  2 ) )
53, 4rabeqbidv 3101 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  { x  e.  ~P v  |  ( # `  x
)  =  2 }  =  { x  e. 
~P V  |  (
# `  x )  =  2 } )
61, 5sseq12d 3526 . 2  |-  ( ( v  =  V  /\  e  =  E )  ->  ( e  C_  { x  e.  ~P v  |  (
# `  x )  =  2 }  <->  E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } ) )
7 ausgra.1 . 2  |-  G  =  { <. v ,  e
>.  |  e  C_  { x  e.  ~P v  |  ( # `  x
)  =  2 } }
86, 7brabga 4754 1  |-  ( ( V  e.  W  /\  E  e.  X )  ->  ( V G E  <-> 
E  C_  { x  e.  ~P V  |  (
# `  x )  =  2 } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811    C_ wss 3469   ~Pcpw 4003   class class class wbr 4440   {copab 4497   ` cfv 5579   2c2 10574   #chash 12360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499
This theorem is referenced by:  ausisusgra  24018
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