Theorem iununir 3759

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 3610	. . . . . 6 |- ( B = (/) -> U. B = U. (/) )
2		uni0 3628	. . . . . 6 |- U. (/) = (/)
3	1, 2	syl6eq 2129	. . . . 5 |- ( B = (/) -> U. B = (/) )
4	3	uneq2d 3126	. . . 4 |- ( B = (/) -> ( A u. U. B ) = ( A u. (/) ) )
5		un0 3278	. . . 4 |- ( A u. (/) ) = A
6	4, 5	syl6eq 2129	. . 3 |- ( B = (/) -> ( A u. U. B ) = A )
7		iuneq1 3691	. . . 4 |- ( B = (/) -> U_ x e. B ( A u. x ) = U_ x e. (/) ( A u. x ) )
8		0iun 3735	. . . 4 |- U_ x e. (/) ( A u. x ) = (/)
9	7, 8	syl6eq 2129	. . 3 |- ( B = (/) -> U_ x e. B ( A u. x ) = (/) )
10	6, 9	eqeq12d 2095	. 2 |- ( B = (/) -> ( ( A u. U. B ) = U_ x e. B ( A u. x ) <-> A = (/) ) )
11	10	biimpcd 157	1 |- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )

Syntax hints:   -> wi 4   = wceq 1284   u. cun 2971   (/)c0 3251   U.cuni 3601   U_ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-uni 3602  df-iun 3680

This theorem is referenced by: (None)