Theorem indi 3743

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

Proof of Theorem indi

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		andi 867	. . . 4
2		elin 3686	. . . . 5
3		elin 3686	. . . . 5
4	2, 3	orbi12i 521	. . . 4
5	1, 4	bitr4i 252	. . 3
6		elun 3644	. . . 4
7	6	anbi2i 694	. . 3
8		elun 3644	. . 3
9	5, 7, 8	3bitr4i 277	. 2
10	9	ineqri 3691	1

Syntax hints:  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  u.cun 3473  i^icin 3474

This theorem is referenced by:  indir  3745  difindi  3751  undisj2  3879  disjssun  3884  difdifdir  3915  disjpr2  4092  diftpsn3  4168  resundi  5292  fresaun  5761  elfiun  7910  unxpwdom  8036  kmlem2  8552  cdainf  8593  ackbij1lem1  8621  ackbij1lem2  8622  ssxr  9675  incexclem  13648  bitsinv1  14092  bitsinvp1  14099  bitsres  14123  paste  19795  unmbl  21948  ovolioo  21978  uniioombllem4  21995  volcn  22015  ellimc2  22281  lhop2  22416  ex-in  25146  eulerpartgbij  28311  asindmre  30102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482