Theorem uncom 3325

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 730	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3322	. . 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 3323	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 710   = wceq 1373   e. wcel 2178   u. cun 3172

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178

This theorem is referenced by:  equncom  3326  uneq2  3329  un12  3339  un23  3340  ssun2  3345  unss2  3352  ssequn2  3354  undir  3431  dif32  3444  undif2ss  3544  uneqdifeqim  3554  prcom  3719  tpass  3739  prprc1  3751  difsnss  3790  exmid1stab  4268  suc0  4476  fununfun  5336  fvun2  5669  fmptpr  5799  fvsnun2  5805  fsnunfv  5808  omv2  6574  phplem2  6975  undifdc  7047  endjusym  7224  fzsuc2  10236  fseq1p1m1  10251  xnn0nnen  10619  ennnfonelem1  12893  setsslid  12998  lgsquadlem2  15670