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Theorem indi 4008
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 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem indi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 andi 947 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
2 elin 3931 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3931 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
42, 3orbi12i 544 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
51, 4bitr4i 267 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
6 elun 3888 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76anbi2i 732 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
8 elun 3888 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
95, 7, 83bitr4i 292 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)))
109ineqri 3941 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 382  wa 383   = wceq 1624  wcel 2131  cun 3705  cin 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-v 3334  df-un 3712  df-in 3714
This theorem is referenced by:  indir  4010  difindi  4016  undisj2  4166  disjssun  4172  difdifdir  4192  disjpr2  4384  disjpr2OLD  4385  diftpsn3OLD  4470  resundi  5560  fresaun  6228  elfiun  8493  unxpwdom  8651  kmlem2  9157  cdainf  9198  ackbij1lem1  9226  ackbij1lem2  9227  ssxr  10291  incexclem  14759  bitsinv1  15358  bitsinvp1  15365  bitsres  15389  paste  21292  unmbl  23497  ovolioo  23528  uniioombllem4  23546  volcn  23566  ellimc2  23832  lhop2  23969  ex-in  27585  eulerpartgbij  30735  poimirlem3  33717  poimirlem15  33729  asindmre  33800  iunrelexp0  38488  sge0resplit  41118  sge0split  41121
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