Theorem iinuni 4414

Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
iinuni

Distinct variable groups:   ,   ,

Proof of Theorem iinuni

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 3003	. . . 4
2		elun 3644	. . . . 5
3	2	ralbii 2888	. . . 4
4		vex 3112	. . . . . 6
5	4	elint2 4293	. . . . 5
6	5	orbi2i 519	. . . 4
7	1, 3, 6	3bitr4ri 278	. . 3
8	7	abbii 2591	. 2
9		df-un 3480	. 2
10		df-iin 4333	. 2
11	8, 9, 10	3eqtr4i 2496	1

Syntax hints:  \/wo 368  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  u.cun 3473  |^|cint 4286  |^|_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-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-ral 2812  df-v 3111  df-un 3480  df-int 4287  df-iin 4333