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Theorem unass 3420
 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 ((AB) ∪ C) = (A ∪ (BC))

Proof of Theorem unass
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elun 3220 . . 3 (x (A ∪ (BC)) ↔ (x A x (BC)))
2 elun 3220 . . . 4 (x (BC) ↔ (x B x C))
32orbi2i 505 . . 3 ((x A x (BC)) ↔ (x A (x B x C)))
4 elun 3220 . . . . 5 (x (AB) ↔ (x A x B))
54orbi1i 506 . . . 4 ((x (AB) x C) ↔ ((x A x B) x C))
6 orass 510 . . . 4 (((x A x B) x C) ↔ (x A (x B x C)))
75, 6bitr2i 241 . . 3 ((x A (x B x C)) ↔ (x (AB) x C))
81, 3, 73bitrri 263 . 2 ((x (AB) x C) ↔ x (A ∪ (BC)))
98uneqri 3406 1 ((AB) ∪ C) = (A ∪ (BC))
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ cun 3207 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214 This theorem is referenced by:  un12  3421  un23  3422  un4  3423  dfif5  3674  qdass  3819  qdassr  3820  ssunpr  3868  addcass  4415
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