Theorem uncom 3365

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 3362	. . 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 3363	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 716   = wceq 1398   e. wcel 2205   u. cun 3211

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217

This theorem is referenced by:  equncom  3366  uneq2  3369  un12  3379  un23  3380  ssun2  3385  unss2  3392  ssequn2  3394  undir  3473  dif32  3486  undif2ss  3587  uneqdifeqim  3597  prcom  3769  tpass  3789  prprc1  3802  difsnss  3842  exmid1stab  4323  suc0  4534  fununfun  5401  fresaunres2disj  5547  fresaunres1disj  5548  fvun2  5746  fmptpr  5878  fvsnun2  5884  fsnunfv  5887  omv2  6700  phplem2  7109  undifdc  7186  endjusym  7389  fzsuc2  10417  fseq1p1m1  10432  xnn0nnen  10803  hashfibclem  11210  ennnfonelem1  13175  setsslid  13280  lgsquadlem2  15968