Theorem iuncom4 4338

Description: Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Assertion

Ref	Expression
iuncom4

Proof of Theorem iuncom4

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2813	. . . . . . 7
2	1	rexbii 2959	. . . . . 6
3		rexcom4 3129	. . . . . 6
4	2, 3	bitri 249	. . . . 5
5		r19.41v 3009	. . . . . 6
6	5	exbii 1667	. . . . 5
7	4, 6	bitri 249	. . . 4
8		eluni2 4253	. . . . 5
9	8	rexbii 2959	. . . 4
10		df-rex 2813	. . . . 5
11		eliun 4335	. . . . . . 7
12	11	anbi1i 695	. . . . . 6
13	12	exbii 1667	. . . . 5
14	10, 13	bitri 249	. . . 4
15	7, 9, 14	3bitr4i 277	. . 3
16		eliun 4335	. . 3
17		eluni2 4253	. . 3
18	15, 16, 17	3bitr4i 277	. 2
19	18	eqriv 2453	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  U.cuni 4249  U_ciun 4330

This theorem is referenced by:  ituniiun  8823  tgidm  19482  txcmplem2  20143

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-rex 2813  df-v 3111  df-uni 4250  df-iun 4332