Theorem uncom 3116

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 679	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3113	. . 3 |- ( x e. ( B u. A ) <-> ( x e. B \/ x e. A ) )
3	1, 2	bitr4i 185	. 2 |- ( ( x e. A \/ x e. B ) <-> x e. ( B u. A ) )
4	3	uneqri 3114	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977

This theorem is referenced by:  equncom  3117  uneq2  3120  un12  3130  un23  3131  ssun2  3136  unss2  3143  ssequn2  3145  undir  3214  dif32  3227  undif2ss  3319  uneqdifeqim  3328  prcom  3468  tpass  3488  prprc1  3500  difsnss  3531  suc0  4166  fvun2  5261  fmptpr  5376  fvsnun2  5382  fsnunfv  5384  omv2  6068  phplem2  6339  fzsuc2  9096  fseq1p1m1  9111