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Theorem unass 3094
 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 3078 . . 3 (x (A ∪ (B𝐶)) ↔ (x A x (B𝐶)))
2 elun 3078 . . . 4 (x (B𝐶) ↔ (x B x 𝐶))
32orbi2i 678 . . 3 ((x A x (B𝐶)) ↔ (x A (x B x 𝐶)))
4 elun 3078 . . . . 5 (x (AB) ↔ (x A x B))
54orbi1i 679 . . . 4 ((x (AB) x 𝐶) ↔ ((x A x B) x 𝐶))
6 orass 683 . . . 4 (((x A x B) x 𝐶) ↔ (x A (x B x 𝐶)))
75, 6bitr2i 174 . . 3 ((x A (x B x 𝐶)) ↔ (x (AB) x 𝐶))
81, 3, 73bitrri 196 . 2 ((x (AB) x 𝐶) ↔ x (A ∪ (B𝐶)))
98uneqri 3079 1 ((AB) ∪ 𝐶) = (A ∪ (B𝐶))
 Colors of variables: wff set class Syntax hints:   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ∪ cun 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916 This theorem is referenced by:  un12  3095  un23  3096  un4  3097  qdass  3458  qdassr  3459  rdgisucinc  5912  oasuc  5983  fzosplitprm1  8860
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