Theorem iuniin 4341

Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iuniin

Distinct variable groups:   ,   ,   ,

Proof of Theorem iuniin

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 2983	. . . 4
2		vex 3112	. . . . . 6
3		eliin 4336	. . . . . 6
4	2, 3	ax-mp 5	. . . . 5
5	4	rexbii 2959	. . . 4
6		eliun 4335	. . . . 5
7	6	ralbii 2888	. . . 4
8	1, 5, 7	3imtr4i 266	. . 3
9		eliun 4335	. . 3
10		eliin 4336	. . . 4
11	2, 10	ax-mp 5	. . 3
12	8, 9, 11	3imtr4i 266	. 2
13	12	ssriv 3507	1

Syntax hints:  <->wb 184  e.wcel 1818  A.wral 2807  E.wrex 2808  cvv 3109  C_wss 3475  U_ciun 4330  |^|_ciin 4331

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-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-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-iun 4332  df-iin 4333