Theorem unass 3129

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 3113	. . 3 |- ( x e. ( A u. ( B u. C ) ) <-> ( x e. A \/ x e. ( B u. C ) ) )
2		elun 3113	. . . 4 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
3	2	orbi2i 711	. . 3 |- ( ( x e. A \/ x e. ( B u. C ) ) <-> ( x e. A \/ ( x e. B \/ x e. C ) ) )
4		elun 3113	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
5	4	orbi1i 712	. . . 4 |- ( ( x e. ( A u. B ) \/ x e. C ) <-> ( ( x e. A \/ x e. B ) \/ x e. C ) )
6		orass 716	. . . 4 |- ( ( ( x e. A \/ x e. B ) \/ x e. C ) <-> ( x e. A \/ ( x e. B \/ x e. C ) ) )
7	5, 6	bitr2i 183	. . 3 |- ( ( x e. A \/ ( x e. B \/ x e. C ) ) <-> ( x e. ( A u. B ) \/ x e. C ) )
8	1, 3, 7	3bitrri 205	. 2 |- ( ( x e. ( A u. B ) \/ x e. C ) <-> x e. ( A u. ( B u. C ) ) )
9	8	uneqri 3114	1 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )

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:  un12  3130  un23  3131  un4  3132  qdass  3489  qdassr  3490  rdgisucinc  5995  oasuc  6067  fzosplitprm1  9243