Theorem uncom 3363

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 736	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3360	. . 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 3361	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 716   = wceq 1398   e. wcel 2203   u. cun 3209

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215

This theorem is referenced by:  equncom  3364  uneq2  3367  un12  3377  un23  3378  ssun2  3383  unss2  3390  ssequn2  3392  undir  3471  dif32  3484  undif2ss  3585  uneqdifeqim  3595  prcom  3767  tpass  3787  prprc1  3800  difsnss  3840  exmid1stab  4321  suc0  4532  fununfun  5399  fresaunres2disj  5545  fresaunres1disj  5546  fvun2  5744  fmptpr  5876  fvsnun2  5882  fsnunfv  5885  omv2  6698  phplem2  7107  undifdc  7184  endjusym  7387  fzsuc2  10413  fseq1p1m1  10428  xnn0nnen  10799  hashfibclem  11206  ennnfonelem1  13158  setsslid  13263  lgsquadlem2  15951