Theorem unass 3364

Description: Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
unass	|- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )

Proof of Theorem unass

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3348	. . 3 |- ( x e. ( A u. ( B u. C ) ) <-> ( x e. A \/ x e. ( B u. C ) ) )
2		elun 3348	. . . 4 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
3	2	orbi2i 769	. . 3 |- ( ( x e. A \/ x e. ( B u. C ) ) <-> ( x e. A \/ ( x e. B \/ x e. C ) ) )
4		elun 3348	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
5	4	orbi1i 770	. . . 4 |- ( ( x e. ( A u. B ) \/ x e. C ) <-> ( ( x e. A \/ x e. B ) \/ x e. C ) )
6		orass 774	. . . 4 |- ( ( ( x e. A \/ x e. B ) \/ x e. C ) <-> ( x e. A \/ ( x e. B \/ x e. C ) ) )
7	5, 6	bitr2i 185	. . 3 |- ( ( x e. A \/ ( x e. B \/ x e. C ) ) <-> ( x e. ( A u. B ) \/ x e. C ) )
8	1, 3, 7	3bitrri 207	. 2 |- ( ( x e. ( A u. B ) \/ x e. C ) <-> x e. ( A u. ( B u. C ) ) )
9	8	uneqri 3349	1 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )

Syntax hints:   \/ wo 715   = wceq 1397   e. wcel 2202   u. cun 3198

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204

This theorem is referenced by:  un12  3365  un23  3366  un4  3367  qdass  3768  qdassr  3769  rdgisucinc  6551  oasuc  6632  unfidisj  7114  undifdc  7116  djuassen  7432  fzosplitpr  10479  fzosplitprm1  10480  hashunlem  11067  prdsvalstrd  13355  plyun0  15462