Theorem uncom 3303

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 3300	. . 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 3301	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 709   = wceq 1364   e. wcel 2164   u. cun 3151

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157

This theorem is referenced by:  equncom  3304  uneq2  3307  un12  3317  un23  3318  ssun2  3323  unss2  3330  ssequn2  3332  undir  3409  dif32  3422  undif2ss  3522  uneqdifeqim  3532  prcom  3694  tpass  3714  prprc1  3726  difsnss  3764  exmid1stab  4237  suc0  4442  fvun2  5624  fmptpr  5750  fvsnun2  5756  fsnunfv  5759  omv2  6518  phplem2  6909  undifdc  6980  endjusym  7155  fzsuc2  10145  fseq1p1m1  10160  xnn0nnen  10508  ennnfonelem1  12564  setsslid  12669