Theorem uncom 3279

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 728	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3276	. . 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 3277	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 708   = wceq 1353   e. wcel 2148   u. cun 3127

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133

This theorem is referenced by:  equncom  3280  uneq2  3283  un12  3293  un23  3294  ssun2  3299  unss2  3306  ssequn2  3308  undir  3385  dif32  3398  undif2ss  3498  uneqdifeqim  3508  prcom  3668  tpass  3688  prprc1  3700  difsnss  3738  exmid1stab  4208  suc0  4411  fvun2  5583  fmptpr  5708  fvsnun2  5714  fsnunfv  5717  omv2  6465  phplem2  6852  undifdc  6922  endjusym  7094  fzsuc2  10076  fseq1p1m1  10091  ennnfonelem1  12402  setsslid  12507