Theorem unass 1615

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

Assertion

Ref	Expression
unass	⊢ ((A ∪ B) ∪ C) = (A ∪ (B ∪ C))

Proof of Theorem unass

Step	Hyp	Ref	Expression
1		orass 218	. . . 4 ⊢ (((x ∈ A ∨ x ∈ B) ∨ x ∈ C) ↔ (x ∈ A ∨ (x ∈ B ∨ x ∈ C)))
2		elun 1601	. . . . 5 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
3	2	orbi1i 215	. . . 4 ⊢ ((x ∈ (A ∪ B) ∨ x ∈ C) ↔ ((x ∈ A ∨ x ∈ B) ∨ x ∈ C))
4		elun 1601	. . . . 5 ⊢ (x ∈ (B ∪ C) ↔ (x ∈ B ∨ x ∈ C))
5	4	orbi2i 214	. . . 4 ⊢ ((x ∈ A ∨ x ∈ (B ∪ C)) ↔ (x ∈ A ∨ (x ∈ B ∨ x ∈ C)))
6	1, 3, 5	3bitr4 158	. . 3 ⊢ ((x ∈ (A ∪ B) ∨ x ∈ C) ↔ (x ∈ A ∨ x ∈ (B ∪ C)))
7		elun 1601	. . 3 ⊢ (x ∈ (A ∪ (B ∪ C)) ↔ (x ∈ A ∨ x ∈ (B ∪ C)))
8	6, 7	bitr4 154	. 2 ⊢ ((x ∈ (A ∪ B) ∨ x ∈ C) ↔ x ∈ (A ∪ (B ∪ C)))
9	8	uneqri 1602	1 ⊢ ((A ∪ B) ∪ C) = (A ∪ (B ∪ C))

Syntax hints:   ∨ wo 195   = wceq 1091   ∈ wcel 1092   ∪ cun 1485

This theorem is referenced by:  un12 1616  un23 1617  un4 1618  cdaassen 3725  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490

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