Theorem uniun 4888

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 1884	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
2		elun 4107	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
3	2	anbi2i 624	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)))
4		andi 1010	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
5	3, 4	bitri 275	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
6	5	exbii 1850	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
7		eluni 4868	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
8		eluni 4868	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
9	7, 8	orbi12i 915	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
10	1, 6, 9	3bitr4i 303	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵))
11		eluni 4868	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
12		elun 4107	. . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵))
13	10, 11, 12	3bitr4i 303	. 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵))
14	13	eqriv 2734	1 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)

Syntax hints:   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ∪ cun 3901  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-uni 4866

This theorem is referenced by:  unidif0  5307  unisucs  6404  fvun  6932  onuninsuci  7792  tc2  9661  fin1a2lem10  10331  fin1a2lem12  10333  incexclem  15771  dprd2da  19985  dmdprdsplit2lem  19988  ordtuni  23146  cmpcld  23358  uncmp  23359  refun0  23471  lfinun  23481  1stckgenlem  23509  filconn  23839  ufildr  23887  alexsubALTlem3  24005  cldsubg  24067  icccmplem2  24780  uniioombllem3  25554  madeoldsuc  27893  sxbrsigalem0  34448  fiunelcarsg  34493  carsgclctunlem1  34494  carsggect  34495  tz9.1regs  35309  cvmscld  35486  refssfne  36571  topjoin  36578  pibt2  37666  mbfresfi  37911  onsucunitp  43724  oaun3  43733  fourierdlem80  46538  isomenndlem  46882