Theorem iununi 4051

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 2519	. . . . . . 7 |- (B =/= (/) <-> -. B = (/))
2		iunconst 3978	. . . . . . 7 |- (B =/= (/) -> U_x e. B A = A)
3	1, 2	sylbir 204	. . . . . 6 |- (-. B = (/) -> U_x e. B A = A)
4		iun0 4023	. . . . . . 7 |- U_x e. B (/) = (/)
5		id 19	. . . . . . . 8 |- (A = (/) -> A = (/))
6	5	iuneq2d 3995	. . . . . . 7 |- (A = (/) -> U_x e. B A = U_x e. B (/))
7	4, 6, 5	3eqtr4a 2411	. . . . . 6 |- (A = (/) -> U_x e. B A = A)
8	3, 7	ja 153	. . . . 5 |- ((B = (/) -> A = (/)) -> U_x e. B A = A)
9	8	eqcomd 2358	. . . 4 |- ((B = (/) -> A = (/)) -> A = U_x e. B A)
10	9	uneq1d 3418	. . 3 |- ((B = (/) -> A = (/)) -> (A u. U_x e. B x) = (U_x e. B A u. U_x e. B x))
11		uniiun 4020	. . . 4 |- U.B = U_x e. B x
12	11	uneq2i 3416	. . 3 |- (A u. U.B) = (A u. U_x e. B x)
13		iunun 4047	. . 3 |- U_x e. B (A u. x) = (U_x e. B A u. U_x e. B x)
14	10, 12, 13	3eqtr4g 2410	. 2 |- ((B = (/) -> A = (/)) -> (A u. U.B) = U_x e. B (A u. x))
15		unieq 3901	. . . . . . 7 |- (B = (/) -> U.B = U.(/))
16		uni0 3919	. . . . . . 7 |- U.(/) = (/)
17	15, 16	syl6eq 2401	. . . . . 6 |- (B = (/) -> U.B = (/))
18	17	uneq2d 3419	. . . . 5 |- (B = (/) -> (A u. U.B) = (A u. (/)))
19		un0 3576	. . . . 5 |- (A u. (/)) = A
20	18, 19	syl6eq 2401	. . . 4 |- (B = (/) -> (A u. U.B) = A)
21		iuneq1 3983	. . . . 5 |- (B = (/) -> U_x e. B (A u. x) = U_x e. (/) (A u. x))
22		0iun 4024	. . . . 5 |- U_x e. (/) (A u. x) = (/)
23	21, 22	syl6eq 2401	. . . 4 |- (B = (/) -> U_x e. B (A u. x) = (/))
24	20, 23	eqeq12d 2367	. . 3 |- (B = (/) -> ((A u. U.B) = U_x e. B (A u. x) <-> A = (/)))
25	24	biimpcd 215	. 2 |- ((A u. U.B) = U_x e. B (A u. x) -> (B = (/) -> A = (/)))
26	14, 25	impbii 180	1 |- ((B = (/) -> A = (/)) <-> (A u. U.B) = U_x e. B (A u. x))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   = wceq 1642   =/= wne 2517   u. cun 3208  (/)c0 3551  U.cuni 3892  U_ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-uni 3893  df-iun 3972

This theorem is referenced by: (None)