Theorem iunxun 5025

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 4127	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2		eliun 4927	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
3		eliun 4927	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
4	2, 3	orbi12i 915	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
5	1, 4	bitr4i 278	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
6		eliun 4927	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		elun 4085	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
8	5, 6, 7	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶))
9	8	eqriv 2732	1 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∃wrex 3059   ∪ cun 3883  ∪ ciun 4923

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 2707

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 2714  df-cleq 2727  df-clel 2810  df-rex 3060  df-v 3429  df-un 3890  df-iun 4925

This theorem is referenced by:  iunxdif3  5026  iunxprg  5027  iunsuc  6399  funiunfv  7192  iunfi  9242  kmlem11  10072  ackbij1lem9  10138  indval2  12153  fsum2dlem  15721  fsumiun  15773  fprod2dlem  15934  prmreclem4  16879  fiuncmp  23357  ovolfiniun  25456  finiunmbl  25499  volfiniun  25502  voliunlem1  25505  uniioombllem4  25541  iuninc  32618  iunxunsn  32624  iunxunpr  32625  ofpreima2  32727  esum2dlem  34224  sigaclfu2  34253  fiunelros  34306  measvuni  34346  cvmliftlem10  35464  mrsubvrs  35692  ttcun  36682  mblfinlem2  37967  dfrcl4  44091  iunrelexp0  44117  comptiunov2i  44121  corclrcl  44122  trclfvdecomr  44143  dfrtrcl4  44153  corcltrcl  44154  cotrclrcl  44157  fiiuncl  45484  iunp1  45485  sge0iunmptlemfi  46829  ovolval4lem1  47065