Theorem uncom 3117

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 680	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3114	. . 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 3115	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 662   = wceq 1285   e. wcel 1434   u. cun 2972

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978

This theorem is referenced by:  equncom  3118  uneq2  3121  un12  3131  un23  3132  ssun2  3137  unss2  3144  ssequn2  3146  undir  3221  dif32  3234  undif2ss  3326  uneqdifeqim  3335  prcom  3476  tpass  3496  prprc1  3508  difsnss  3539  suc0  4174  fvun2  5272  fmptpr  5387  fvsnun2  5393  fsnunfv  5395  omv2  6109  phplem2  6388  undiffi  6443  fzsuc2  9172  fseq1p1m1  9187