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Theorem unass 3128
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 3112 . . 3  |-  ( x  e.  ( A  u.  ( B  u.  C
) )  <->  ( x  e.  A  \/  x  e.  ( B  u.  C
) ) )
2 elun 3112 . . . 4  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
32orbi2i 689 . . 3  |-  ( ( x  e.  A  \/  x  e.  ( B  u.  C ) )  <->  ( x  e.  A  \/  (
x  e.  B  \/  x  e.  C )
) )
4 elun 3112 . . . . 5  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
54orbi1i 690 . . . 4  |-  ( ( x  e.  ( A  u.  B )  \/  x  e.  C )  <-> 
( ( x  e.  A  \/  x  e.  B )  \/  x  e.  C ) )
6 orass 694 . . . 4  |-  ( ( ( x  e.  A  \/  x  e.  B
)  \/  x  e.  C )  <->  ( x  e.  A  \/  (
x  e.  B  \/  x  e.  C )
) )
75, 6bitr2i 178 . . 3  |-  ( ( x  e.  A  \/  ( x  e.  B  \/  x  e.  C
) )  <->  ( x  e.  ( A  u.  B
)  \/  x  e.  C ) )
81, 3, 73bitrri 200 . 2  |-  ( ( x  e.  ( A  u.  B )  \/  x  e.  C )  <-> 
x  e.  ( A  u.  ( B  u.  C ) ) )
98uneqri 3113 1  |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C )
)
Colors of variables: wff set class
Syntax hints:    \/ wo 639    = wceq 1259    e. wcel 1409    u. cun 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950
This theorem is referenced by:  un12  3129  un23  3130  un4  3131  qdass  3495  qdassr  3496  rdgisucinc  6003  oasuc  6075  fzosplitprm1  9192
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