Theorem iunxun 3952

Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

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

Proof of Theorem iunxun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexun 3307	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2		eliun 3877	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
3		eliun 3877	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
4	2, 3	orbi12i 759	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
5	1, 4	bitr4i 186	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
6		eliun 3877	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		elun 3268	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
8	5, 6, 7	3bitr4i 211	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶))
9	8	eqriv 2167	1 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   ∨ wo 703   = wceq 1348   ∈ wcel 2141  ∃wrex 2449   ∪ cun 3119  ∪ ciun 3873

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-iun 3875

This theorem is referenced by:  iunxprg  3953  iunsuc  4405  rdgisuc1  6363  oasuc  6443  omsuc  6451  iunfidisj  6923  fsum2dlemstep  11397  fsumiun  11440  fprod2dlemstep  11585  iuncld  12909