Theorem unass 3228

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 3212	. . 3 |- ( x e. ( A u. ( B u. C ) ) <-> ( x e. A \/ x e. ( B u. C ) ) )
2		elun 3212	. . . 4 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
3	2	orbi2i 751	. . 3 |- ( ( x e. A \/ x e. ( B u. C ) ) <-> ( x e. A \/ ( x e. B \/ x e. C ) ) )
4		elun 3212	. . . . 5 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
5	4	orbi1i 752	. . . 4 |- ( ( x e. ( A u. B ) \/ x e. C ) <-> ( ( x e. A \/ x e. B ) \/ x e. C ) )
6		orass 756	. . . 4 |- ( ( ( x e. A \/ x e. B ) \/ x e. C ) <-> ( x e. A \/ ( x e. B \/ x e. C ) ) )
7	5, 6	bitr2i 184	. . 3 |- ( ( x e. A \/ ( x e. B \/ x e. C ) ) <-> ( x e. ( A u. B ) \/ x e. C ) )
8	1, 3, 7	3bitrri 206	. 2 |- ( ( x e. ( A u. B ) \/ x e. C ) <-> x e. ( A u. ( B u. C ) ) )
9	8	uneqri 3213	1 |- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )

Syntax hints:   \/ wo 697   = wceq 1331   e. wcel 1480   u. cun 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070

This theorem is referenced by:  un12  3229  un23  3230  un4  3231  qdass  3615  qdassr  3616  rdgisucinc  6275  oasuc  6353  unfidisj  6803  undifdc  6805  djuassen  7066  fzosplitprm1  10004  hashunlem  10543