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Theorem unass 3496
Description: Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
unass  |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C )
)

Proof of Theorem unass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elun 3480 . . 3  |-  ( x  e.  ( A  u.  ( B  u.  C
) )  <->  ( x  e.  A  \/  x  e.  ( B  u.  C
) ) )
2 elun 3480 . . . 4  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
32orbi2i 506 . . 3  |-  ( ( x  e.  A  \/  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  \/  (
x  e.  B  \/  x  e.  C )
) )
4 elun 3480 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
54orbi1i 507 . . . 4  |-  ( ( x  e.  ( A  u.  B )  \/  x  e.  C )  <-> 
( ( x  e.  A  \/  x  e.  B )  \/  x  e.  C ) )
6 orass 511 . . . 4  |-  ( ( ( x  e.  A  \/  x  e.  B
)  \/  x  e.  C )  <->  ( x  e.  A  \/  (
x  e.  B  \/  x  e.  C )
) )
75, 6bitr2i 242 . . 3  |-  ( ( x  e.  A  \/  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  ( A  u.  B
)  \/  x  e.  C ) )
81, 3, 73bitrri 264 . 2  |-  ( ( x  e.  ( A  u.  B )  \/  x  e.  C )  <-> 
x  e.  ( A  u.  ( B  u.  C ) ) )
98uneqri 3481 1  |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C )
)
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1652    e. wcel 1725    u. cun 3310
This theorem is referenced by:  un12  3497  un23  3498  un4  3499  dfif5  3743  qdass  3895  qdassr  3896  ssunpr  3953  oarec  6796  domunfican  7370  cdaassen  8051  prunioo  11014  ioojoin  11016  s4prop  11850  strlemor2  13545  strlemor3  13546  phlstr  13596  prdsvalstr  13664  mreexexlem2d  13858  mreexexlem4d  13860  ordtbas  17244  reconnlem1  18845  lhop  19888  plyun0  20104  ex-un  21720  ex-pw  21725  subfacp1lem1  24853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317
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