Theorem uncom 3261

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 718	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3258	. . 3 |- ( x e. ( B u. A ) <-> ( x e. B \/ x e. A ) )
3	1, 2	bitr4i 186	. 2 |- ( ( x e. A \/ x e. B ) <-> x e. ( B u. A ) )
4	3	uneqri 3259	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 698   = wceq 1342   e. wcel 2135   u. cun 3109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115

This theorem is referenced by:  equncom  3262  uneq2  3265  un12  3275  un23  3276  ssun2  3281  unss2  3288  ssequn2  3290  undir  3367  dif32  3380  undif2ss  3479  uneqdifeqim  3489  prcom  3646  tpass  3666  prprc1  3678  difsnss  3713  suc0  4383  fvun2  5547  fmptpr  5671  fvsnun2  5677  fsnunfv  5680  omv2  6424  phplem2  6810  undifdc  6880  endjusym  7052  fzsuc2  10004  fseq1p1m1  10019  ennnfonelem1  12277  setsslid  12381  exmid1stab  13714