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Theorem isausgra 24771
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 459 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  e  =  E )
2 pweq 3958 . . . . 5  |-  ( v  =  V  ->  ~P v  =  ~P V
)
32adantr 463 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ~P v  =  ~P V )
4 biidd 237 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( # `  x
)  =  2  <->  ( # `
 x )  =  2 ) )
53, 4rabeqbidv 3054 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  { x  e.  ~P v  |  ( # `  x
)  =  2 }  =  { x  e. 
~P V  |  (
# `  x )  =  2 } )
61, 5sseq12d 3471 . 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 4704 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 367    = wceq 1405    e. wcel 1842   {crab 2758    C_ wss 3414   ~Pcpw 3955   class class class wbr 4395   {copab 4452   ` cfv 5569   2c2 10626   #chash 12452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454
This theorem is referenced by:  ausisusgra  24772
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