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Theorem uncom 3224
Description: Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
uncom  |-  ( A  u.  B )  =  ( B  u.  A
)

Proof of Theorem uncom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 orcom 718 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  ( x  e.  B  \/  x  e.  A )
)
2 elun 3221 . . 3  |-  ( x  e.  ( B  u.  A )  <->  ( x  e.  B  \/  x  e.  A ) )
31, 2bitr4i 186 . 2  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  ( B  u.  A ) )
43uneqri 3222 1  |-  ( A  u.  B )  =  ( B  u.  A
)
Colors of variables: wff set class
Syntax hints:    \/ wo 698    = wceq 1332    e. wcel 1481    u. cun 3073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079
This theorem is referenced by:  equncom  3225  uneq2  3228  un12  3238  un23  3239  ssun2  3244  unss2  3251  ssequn2  3253  undir  3330  dif32  3343  undif2ss  3442  uneqdifeqim  3452  prcom  3606  tpass  3626  prprc1  3638  difsnss  3673  suc0  4340  fvun2  5495  fmptpr  5619  fvsnun2  5625  fsnunfv  5628  omv2  6368  phplem2  6754  undifdc  6819  endjusym  6988  fzsuc2  9889  fseq1p1m1  9904  ennnfonelem1  11954  setsslid  12046  exmid1stab  13366
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