Theorem iununi 4610

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	|- ( ( B = (/) -> A = (/) ) <-> ( A u. U. B ) = U_ x e. B ( A u. x ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem iununi

Step	Hyp	Ref	Expression
1		df-ne 2795	. . . . . . 7 |- ( B =/= (/) <-> -. B = (/) )
2		iunconst 4529	. . . . . . 7 |- ( B =/= (/) -> U_ x e. B A = A )
3	1, 2	sylbir 225	. . . . . 6 |- ( -. B = (/) -> U_ x e. B A = A )
4		iun0 4576	. . . . . . 7 |- U_ x e. B (/) = (/)
5		id 22	. . . . . . . 8 |- ( A = (/) -> A = (/) )
6	5	iuneq2d 4547	. . . . . . 7 |- ( A = (/) -> U_ x e. B A = U_ x e. B (/) )
7	4, 6, 5	3eqtr4a 2682	. . . . . 6 |- ( A = (/) -> U_ x e. B A = A )
8	3, 7	ja 173	. . . . 5 |- ( ( B = (/) -> A = (/) ) -> U_ x e. B A = A )
9	8	eqcomd 2628	. . . 4 |- ( ( B = (/) -> A = (/) ) -> A = U_ x e. B A )
10	9	uneq1d 3766	. . 3 |- ( ( B = (/) -> A = (/) ) -> ( A u. U_ x e. B x ) = ( U_ x e. B A u. U_ x e. B x ) )
11		uniiun 4573	. . . 4 |- U. B = U_ x e. B x
12	11	uneq2i 3764	. . 3 |- ( A u. U. B ) = ( A u. U_ x e. B x )
13		iunun 4604	. . 3 |- U_ x e. B ( A u. x ) = ( U_ x e. B A u. U_ x e. B x )
14	10, 12, 13	3eqtr4g 2681	. 2 |- ( ( B = (/) -> A = (/) ) -> ( A u. U. B ) = U_ x e. B ( A u. x ) )
15		unieq 4444	. . . . . . 7 |- ( B = (/) -> U. B = U. (/) )
16		uni0 4465	. . . . . . 7 |- U. (/) = (/)
17	15, 16	syl6eq 2672	. . . . . 6 |- ( B = (/) -> U. B = (/) )
18	17	uneq2d 3767	. . . . 5 |- ( B = (/) -> ( A u. U. B ) = ( A u. (/) ) )
19		un0 3967	. . . . 5 |- ( A u. (/) ) = A
20	18, 19	syl6eq 2672	. . . 4 |- ( B = (/) -> ( A u. U. B ) = A )
21		iuneq1 4534	. . . . 5 |- ( B = (/) -> U_ x e. B ( A u. x ) = U_ x e. (/) ( A u. x ) )
22		0iun 4577	. . . . 5 |- U_ x e. (/) ( A u. x ) = (/)
23	21, 22	syl6eq 2672	. . . 4 |- ( B = (/) -> U_ x e. B ( A u. x ) = (/) )
24	20, 23	eqeq12d 2637	. . 3 |- ( B = (/) -> ( ( A u. U. B ) = U_ x e. B ( A u. x ) <-> A = (/) ) )
25	24	biimpcd 239	. 2 |- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )
26	14, 25	impbii 199	1 |- ( ( B = (/) -> A = (/) ) <-> ( A u. U. B ) = U_ x e. B ( A u. x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   =/= wne 2794   u. cun 3572   (/)c0 3915   U.cuni 4436   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437  df-iun 4522

This theorem is referenced by: (None)