Theorem uncom 3373

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.

Step	Hyp	Ref	Expression
1		orcom 740	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3370	. . 3 |- ( x e. ( B u. A ) <-> ( x e. B \/ x e. A ) )
3	1, 2	bitr4i 187	. 2 |- ( ( x e. A \/ x e. B ) <-> x e. ( B u. A ) )
4	3	uneqri 3371	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 720   = wceq 1402   e. wcel 2209   u. cun 3218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is referenced by:  equncom  3374  uneq2  3377  un12  3387  un23  3388  ssun2  3393  unss2  3400  ssequn2  3402  undir  3481  dif32  3494  undif2ss  3600  uneqdifeqim  3610  prcom  3783  tpass  3803  prprc1  3816  difsnss  3856  exmid1stab  4340  suc0  4551  fununfun  5419  fresaunres2disj  5565  fresaunres1disj  5566  fvun2  5764  fmptpr  5898  fvsnun2  5904  fsnunfv  5907  omv2  6728  phplem2  7144  undifdc  7221  endjusym  7426  fzsuc2  10464  fseq1p1m1  10479  xnn0nnen  10852  hashfibclem  11260  ennnfonelem1  13276  setsslid  13381  lgsquadlem2  16111