Theorem uncom 3308

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 729	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3305	. . 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 3306	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 709   = wceq 1364   e. wcel 2167   u. cun 3155

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161

This theorem is referenced by:  equncom  3309  uneq2  3312  un12  3322  un23  3323  ssun2  3328  unss2  3335  ssequn2  3337  undir  3414  dif32  3427  undif2ss  3527  uneqdifeqim  3537  prcom  3699  tpass  3719  prprc1  3731  difsnss  3769  exmid1stab  4242  suc0  4447  fvun2  5631  fmptpr  5757  fvsnun2  5763  fsnunfv  5766  omv2  6532  phplem2  6923  undifdc  6994  endjusym  7171  fzsuc2  10171  fseq1p1m1  10186  xnn0nnen  10546  ennnfonelem1  12649  setsslid  12754  lgsquadlem2  15403