Theorem indi 4229

Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
indi	⊢ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶))

Proof of Theorem indi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		andi 1025	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)))
2		elin 4093	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
3		elin 4093	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶))
4	2, 3	orbi12i 928	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∩ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)))
5	1, 4	bitr4i 281	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∩ 𝐶)))
6		elun 4152	. . . 4 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
7	6	anbi2i 635	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
8		elun 4152	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∩ 𝐶)))
9	5, 7, 8	3bitr4i 306	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)))
10	9	ineqri 4109	1 ⊢ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶))

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ∩ cin 4089   ∪ cun 4149

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-v 3452  df-in 4091  df-un 4151

This theorem is used by:  indir  4231  difindi  4237  undisj2  4415  disjssun  4420  difdifdir  4446  disjpr2  4673  resundi  5983  fresaun  6753  elfiun  9422  unxpwdom  9583  kmlem2  10230  djuinf  10267  ackbij1lem1  10297  ackbij1lem2  10298  ssxr  11379  incexclem  16005  bitsinv1  16612  bitsinvp1  16619  bitsres  16643  paste  23612  unmbl  25858  ovolioo  25889  uniioombllem4  25907  volcn  25927  ellimc2  26197  lhop2  26335  ex-in  31026  eulerpartgbij  35004  poimirlem3  38542  poimirlem15  38554  asindmre  38622  iunrelexp0  44702  sge0resplit  47416  sge0split  47419  tposresg  49986  tposrescnv  49987  iscnrm3rlem1  50048  topdlat  50112