Theorem iununi 4415

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

Assertion

Ref	Expression
iununi

Distinct variable groups:   ,   ,

Proof of Theorem iununi

Step	Hyp	Ref	Expression
1		df-ne 2654	. . . . . . 7
2		iunconst 4339	. . . . . . 7
3	1, 2	sylbir 213	. . . . . 6
4		iun0 4386	. . . . . . 7
5		id 22	. . . . . . . 8
6	5	iuneq2d 4357	. . . . . . 7
7	4, 6, 5	3eqtr4a 2524	. . . . . 6
8	3, 7	ja 161	. . . . 5
9	8	eqcomd 2465	. . . 4
10	9	uneq1d 3656	. . 3
11		uniiun 4383	. . . 4
12	11	uneq2i 3654	. . 3
13		iunun 4411	. . 3
14	10, 12, 13	3eqtr4g 2523	. 2
15		unieq 4257	. . . . . . 7
16		uni0 4276	. . . . . . 7
17	15, 16	syl6eq 2514	. . . . . 6
18	17	uneq2d 3657	. . . . 5
19		un0 3810	. . . . 5
20	18, 19	syl6eq 2514	. . . 4
21		iuneq1 4344	. . . . 5
22		0iun 4387	. . . . 5
23	21, 22	syl6eq 2514	. . . 4
24	20, 23	eqeq12d 2479	. . 3
25	24	biimpcd 224	. 2
26	14, 25	impbii 188	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  =wceq 1395  =/=wne 2652  u.cun 3473  c0 3784  U.cuni 4249  U_ciun 4330

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-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250  df-iun 4332