Theorem iunxun 4021

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 3361	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2		eliun 3945	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
3		eliun 3945	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
4	2, 3	orbi12i 766	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
5	1, 4	bitr4i 187	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
6		eliun 3945	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		elun 3322	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
8	5, 6, 7	3bitr4i 212	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶))
9	8	eqriv 2204	1 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 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:  iunxprg  4022  iunsuc  4485  rdgisuc1  6493  oasuc  6573  omsuc  6581  iunfidisj  7074  fsum2dlemstep  11860  fsumiun  11903  fprod2dlemstep  12048  iuncld  14702