Theorem uniun 3871

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 1652	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
2		elun 3315	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
3	2	anbi2i 457	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)))
4		andi 820	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
5	3, 4	bitri 184	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
6	5	exbii 1629	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ ∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
7		eluni 3855	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
8		eluni 3855	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
9	7, 8	orbi12i 766	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∨ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
10	1, 6, 9	3bitr4i 212	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵))
11		eluni 3855	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∪ 𝐵)))
12		elun 3315	. . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵))
13	10, 11, 12	3bitr4i 212	. 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ (∪ 𝐴 ∪ ∪ 𝐵))
14	13	eqriv 2203	1 ⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵)

Syntax hints:   ∧ wa 104   ∨ wo 710   = wceq 1373  ∃wex 1516   ∈ wcel 2177   ∪ cun 3165  ∪ cuni 3852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3171  df-uni 3853

This theorem is referenced by:  unisuc  4464  unisucg  4465