Theorem uniun 3763

Description: The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)

Assertion

Ref	Expression
uniun	⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)

Proof of Theorem uniun

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.43 1608	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
2		elun 3222	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
3	2	anbi2i 453	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)))
4		andi 808	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
5	3, 4	bitri 183	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
6	5	exbii 1585	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
7		eluni 3747	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
8		eluni 3747	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
9	7, 8	orbi12i 754	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
10	1, 6, 9	3bitr4i 211	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵))
11		eluni 3747	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
12		elun 3222	. . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵))
13	10, 11, 12	3bitr4i 211	. 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵))
14	13	eqriv 2137	1 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)

Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481   ∪ cun 3074  ∪ cuni 3744

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-uni 3745

This theorem is referenced by:  unisuc  4343  unisucg  4344