Theorem iunun 4020

Description: Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
iunun	⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem iunun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.43 2666	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∨ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		elun 3322	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
3	2	rexbii 2515	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
4		eliun 3945	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
5		eliun 3945	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6	4, 5	orbi12i 766	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∨ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
7	1, 3, 6	3bitr4i 212	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
8		eliun 3945	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶))
9		elun 3322	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
10	7, 8, 9	3bitr4i 212	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶))
11	10	eqriv 2204	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ∨ wo 710   = wceq 1373   ∈ wcel 2178  ∃wrex 2487   ∪ cun 3172  ∪ ciun 3941

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-iun 3943

This theorem is referenced by: (None)