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Theorem isausgra 23415
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 3963 . . . . 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 3065 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  { x  e.  ~P v  |  ( # `  x
)  =  2 }  =  { x  e. 
~P V  |  (
# `  x )  =  2 } )
61, 5sseq12d 3485 . 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 4703 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 1370    e. wcel 1758   {crab 2799    C_ wss 3428   ~Pcpw 3960   class class class wbr 4392   {copab 4449   ` cfv 5518   2c2 10474   #chash 12206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451
This theorem is referenced by:  ausisusgra  23416
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