Theorem iinin1 4401

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4384 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
iinin1

Distinct variable groups:   ,   ,

Proof of Theorem iinin1

Step	Hyp	Ref	Expression
1		iinin2 4400	. 2
2		incom 3690	. . . 4
3	2	a1i 11	. . 3
4	3	iineq2i 4350	. 2
5		incom 3690	. 2
6	1, 4, 5	3eqtr4g 2523	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  =/=wne 2652  i^icin 3474  c0 3784  |^|_ciin 4331

This theorem is referenced by:  firest  14830

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-ne 2654  df-ral 2812  df-v 3111  df-dif 3478  df-in 3482  df-nul 3785  df-iin 4333