Theorem iunxun 5053

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 4194	. . . 4 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
2		eliun 4954	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
3		eliun 4954	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
4	2, 3	orbi12i 928	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ∨ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
5	1, 4	bitr4i 281	. . 3 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶 ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
6		eliun 4954	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 ∪ 𝐵)𝑦 ∈ 𝐶)
7		elun 4152	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ∨ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶))
8	5, 6, 7	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ 𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶))
9	8	eqriv 2757	1 ⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶)

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ∪ cun 4149  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452  df-un 4151  df-iun 4952

This theorem is used by:  iunxdif3  5054  iunxprg  5055  iunsuc  6450  funiunfv  7252  iunfi  9332  kmlem11  10239  ackbij1lem9  10305  indval2  12325  fsum2dlem  15936  fsumiun  15988  fprod2dlem  16147  prmreclem4  17097  fiuncmp  23722  ovolfiniun  25822  finiunmbl  25865  volfiniun  25868  voliunlem1  25871  uniioombllem4  25907  iuninc  33155  iunxunsn  33160  iunxunpr  33161  ofpreima2  33260  esum2dlem  34724  sigaclfu2  34753  fiunelros  34807  measvuni  34847  cvmliftlem10  36059  mrsubvrs  36287  ttcun  37300  mblfinlem2  38577  dfrcl4  44676  iunrelexp0  44702  comptiunov2i  44706  corclrcl  44707  trclfvdecomr  44728  dfrtrcl4  44738  corcltrcl  44739  cotrclrcl  44742  fiiuncl  46082  iunp1  46083  sge0iunmptlemfi  47423  ovolval4lem1  47659