Theorem uncom 3184

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 700	. . 3 |- ( ( x e. A \/ x e. B ) <-> ( x e. B \/ x e. A ) )
2		elun 3181	. . 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 3182	1 |- ( A u. B ) = ( B u. A )

Syntax hints:   \/ wo 680   = wceq 1312   e. wcel 1461   u. cun 3033

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039

This theorem is referenced by:  equncom  3185  uneq2  3188  un12  3198  un23  3199  ssun2  3204  unss2  3211  ssequn2  3213  undir  3290  dif32  3303  undif2ss  3402  uneqdifeqim  3412  prcom  3563  tpass  3583  prprc1  3595  difsnss  3630  suc0  4291  fvun2  5440  fmptpr  5564  fvsnun2  5570  fsnunfv  5573  omv2  6313  phplem2  6698  undifdc  6763  endjusym  6931  fzsuc2  9746  fseq1p1m1  9761  ennnfonelem1  11759  setsslid  11846