Theorem iununir 3738

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 3589	. . . . . 6 |- ( B = (/) -> U. B = U. (/) )
2		uni0 3607	. . . . . 6 |- U. (/) = (/)
3	1, 2	syl6eq 2088	. . . . 5 |- ( B = (/) -> U. B = (/) )
4	3	uneq2d 3097	. . . 4 |- ( B = (/) -> ( A u. U. B ) = ( A u. (/) ) )
5		un0 3251	. . . 4 |- ( A u. (/) ) = A
6	4, 5	syl6eq 2088	. . 3 |- ( B = (/) -> ( A u. U. B ) = A )
7		iuneq1 3670	. . . 4 |- ( B = (/) -> U_ x e. B ( A u. x ) = U_ x e. (/) ( A u. x ) )
8		0iun 3714	. . . 4 |- U_ x e. (/) ( A u. x ) = (/)
9	7, 8	syl6eq 2088	. . 3 |- ( B = (/) -> U_ x e. B ( A u. x ) = (/) )
10	6, 9	eqeq12d 2054	. 2 |- ( B = (/) -> ( ( A u. U. B ) = U_ x e. B ( A u. x ) <-> A = (/) ) )
11	10	biimpcd 148	1 |- ( ( A u. U. B ) = U_ x e. B ( A u. x ) -> ( B = (/) -> A = (/) ) )

Syntax hints:   -> wi 4   = wceq 1243   u. cun 2915   (/)c0 3224   U.cuni 3580   U_ciun 3657

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581  df-iun 3659

This theorem is referenced by: (None)