Theorem iununir 4080

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 3928	. . . . . 6 |- ( B = (/) -> U. B = U. (/) )
2		uni0 3946	. . . . . 6 |- U. (/) = (/)
3	1, 2	eqtrdi 2283	. . . . 5 |- ( B = (/) -> U. B = (/) )
4	3	uneq2d 3377	. . . 4 |- ( B = (/) -> ( A u. U. B ) = ( A u. (/) ) )
5		un0 3546	. . . 4 |- ( A u. (/) ) = A
6	4, 5	eqtrdi 2283	. . 3 |- ( B = (/) -> ( A u. U. B ) = A )
7		iuneq1 4009	. . . 4 |- ( B = (/) -> U_ x e. B ( A u. x ) = U_ x e. (/) ( A u. x ) )
8		0iun 4054	. . . 4 |- U_ x e. (/) ( A u. x ) = (/)
9	7, 8	eqtrdi 2283	. . 3 |- ( B = (/) -> U_ x e. B ( A u. x ) = (/) )
10	6, 9	eqeq12d 2249	. 2 |- ( B = (/) -> ( ( A u. U. B ) = U_ x e. B ( A u. x ) <-> A = (/) ) )
11	10	biimpcd 159	1 |- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3212   (/)c0 3512   U.cuni 3919   U_ciun 3996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-sn 3700  df-uni 3920  df-iun 3998

This theorem is referenced by: (None)