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Theorem uncom 3133
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 680 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  ( x  e.  B  \/  x  e.  A )
)
2 elun 3130 . . 3  |-  ( x  e.  ( B  u.  A )  <->  ( x  e.  B  \/  x  e.  A ) )
31, 2bitr4i 185 . 2  |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  ( B  u.  A ) )
43uneqri 3131 1  |-  ( A  u.  B )  =  ( B  u.  A
)
Colors of variables: wff set class
Syntax hints:    \/ wo 662    = wceq 1287    e. wcel 1436    u. cun 2986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992
This theorem is referenced by:  equncom  3134  uneq2  3137  un12  3147  un23  3148  ssun2  3153  unss2  3160  ssequn2  3162  undir  3238  dif32  3251  undif2ss  3346  uneqdifeqim  3355  prcom  3501  tpass  3521  prprc1  3533  difsnss  3566  suc0  4212  fvun2  5334  fmptpr  5452  fvsnun2  5458  fsnunfv  5460  omv2  6180  phplem2  6521  undifdc  6586  fzsuc2  9423  fseq1p1m1  9438
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