Theorem uncom 3373

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 740	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3370	. . 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 3371	1 |- ( A u. B ) = ( B u. A )

This proof depends on syntax axioms:   \/ wo 720   = wceq 1402   e. wcel 2209   u. cun 3218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is used by:  equncom  3374  uneq2  3377  un12  3387  un23  3388  ssun2  3393  unss2  3400  ssequn2  3402  undir  3481  dif32  3494  undif2ss  3603  uneqdifeqim  3613  prcom  3787  tpass  3807  prprc1  3821  difsnss  3861  exmid1stab  4347  suc0  4558  fununfun  5427  fresaunres2disj  5573  fresaunres1disj  5574  fvun2  5773  fmptpr  5910  fvsnun2  5916  fsnunfv  5919  omv2  6742  phplem2  7158  undifdc  7235  endjusym  7442  fzsuc2  10503  fseq1p1m1  10518  xnn0nnen  10895  hashfibclem  11304  ennnfonelem1  13356  setsslid  13461  gsumsnfdcmn  14257  birthdaylem2  16226  lgsquadlem2  16402