Theorem unass 2187

Description: Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17.

Assertion

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

Proof of Theorem unass

Step	Hyp	Ref	Expression
1		orass 260	. . . 4 |- (((x e. A \/ x e. B) \/ x e. C) <-> (x e. A \/ (x e. B \/ x e. C)))
2		elun 2173	. . . . 5 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
3	2	orbi1i 256	. . . 4 |- ((x e. (A u. B) \/ x e. C) <-> ((x e. A \/ x e. B) \/ x e. C))
4		elun 2173	. . . . 5 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
5	4	orbi2i 255	. . . 4 |- ((x e. A \/ x e. (B u. C)) <-> (x e. A \/ (x e. B \/ x e. C)))
6	1, 3, 5	3bitr4 183	. . 3 |- ((x e. (A u. B) \/ x e. C) <-> (x e. A \/ x e. (B u. C)))
7		elun 2173	. . 3 |- (x e. (A u. (B u. C)) <-> (x e. A \/ x e. (B u. C)))
8	6, 7	bitr4 176	. 2 |- ((x e. (A u. B) \/ x e. C) <-> x e. (A u. (B u. C)))
9	8	uneqri 2174	1 |- ((A u. B) u. C) = (A u. (B u. C))

Syntax hints:   \/ wo 222   = wceq 956   e. wcel 958   u. cun 2045

This theorem is referenced by:  un12 2188  un23 2189  un4 2190  oarec 4196  cdaassen 4930  ioojoint 6416  infxpidmlem11 7562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050

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