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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
Assertion
Ref Expression
isausgra
Distinct variable groups:   ,,,   ,,,   ,
Allowed substitution hints:   (,,)   (,,)   (,)

Proof of Theorem isausgra
StepHypRef Expression
1 simpr 459 . . 3
2 pweq 3958 . . . . 5
32adantr 463 . . . 4
4 biidd 237 . . . 4
53, 4rabeqbidv 3054 . . 3
61, 5sseq12d 3471 . 2
7 ausgra.1 . 2
86, 7brabga 4704 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405   wcel 1842  crab 2758   wss 3414  cpw 3955   class class class wbr 4395  copab 4452  cfv 5569  c2 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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