Theorem iununir 4000

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 3848	. . . . . 6 |- ( B = (/) -> U. B = U. (/) )
2		uni0 3866	. . . . . 6 |- U. (/) = (/)
3	1, 2	eqtrdi 2245	. . . . 5 |- ( B = (/) -> U. B = (/) )
4	3	uneq2d 3317	. . . 4 |- ( B = (/) -> ( A u. U. B ) = ( A u. (/) ) )
5		un0 3484	. . . 4 |- ( A u. (/) ) = A
6	4, 5	eqtrdi 2245	. . 3 |- ( B = (/) -> ( A u. U. B ) = A )
7		iuneq1 3929	. . . 4 |- ( B = (/) -> U_ x e. B ( A u. x ) = U_ x e. (/) ( A u. x ) )
8		0iun 3974	. . . 4 |- U_ x e. (/) ( A u. x ) = (/)
9	7, 8	eqtrdi 2245	. . 3 |- ( B = (/) -> U_ x e. B ( A u. x ) = (/) )
10	6, 9	eqeq12d 2211	. 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 1364   u. cun 3155   (/)c0 3450   U.cuni 3839   U_ciun 3916

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-uni 3840  df-iun 3918

This theorem is referenced by: (None)