Theorem uncom 3351

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 735	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3348	. . 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 3349	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 715   = wceq 1397   e. wcel 2202   u. cun 3198

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204

This theorem is referenced by:  equncom  3352  uneq2  3355  un12  3365  un23  3366  ssun2  3371  unss2  3378  ssequn2  3380  undir  3457  dif32  3470  undif2ss  3570  uneqdifeqim  3580  prcom  3747  tpass  3767  prprc1  3780  difsnss  3819  exmid1stab  4298  suc0  4508  fununfun  5373  fvun2  5713  fmptpr  5845  fvsnun2  5851  fsnunfv  5854  omv2  6632  phplem2  7038  undifdc  7115  endjusym  7294  fzsuc2  10313  fseq1p1m1  10328  xnn0nnen  10698  ennnfonelem1  13027  setsslid  13132  lgsquadlem2  15806