Theorem iununir 3896

Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
iununir	|- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem iununir

Step	Hyp	Ref	Expression
1		unieq 3745	. . . . . 6 |- ( B = (/) -> U. B = U. (/) )
2		uni0 3763	. . . . . 6 |- U. (/) = (/)
3	1, 2	syl6eq 2188	. . . . 5 |- ( B = (/) -> U. B = (/) )
4	3	uneq2d 3230	. . . 4 |- ( B = (/) -> ( A u. U. B ) = ( A u. (/) ) )
5		un0 3396	. . . 4 |- ( A u. (/) ) = A
6	4, 5	syl6eq 2188	. . 3 |- ( B = (/) -> ( A u. U. B ) = A )
7		iuneq1 3826	. . . 4 |- ( B = (/) -> U_ x e. B ( A u. x ) = U_ x e. (/) ( A u. x ) )
8		0iun 3870	. . . 4 |- U_ x e. (/) ( A u. x ) = (/)
9	7, 8	syl6eq 2188	. . 3 |- ( B = (/) -> U_ x e. B ( A u. x ) = (/) )
10	6, 9	eqeq12d 2154	. 2 |- ( B = (/) -> ( ( A u. U. B ) = U_ x e. B ( A u. x ) <-> A = (/) ) )
11	10	biimpcd 158	1 |- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )

Syntax hints:   -> wi 4   = wceq 1331   u. cun 3069   (/)c0 3363   U.cuni 3736   U_ciun 3813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-uni 3737  df-iun 3815

This theorem is referenced by: (None)