Theorem iuniin 4531

Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iuniin	|- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C

Distinct variable groups:   x, y   y, A   x, B

Allowed substitution hints:   A( x)   B( y)   C( x, y)

Proof of Theorem iuniin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 3063	. . . 4 |- ( E. x e. A A. y e. B z e. C -> A. y e. B E. x e. A z e. C )
2		vex 3203	. . . . . 6 |- z e. _V
3		eliin 4525	. . . . . 6 |- ( z e. _V -> ( z e. |^|_ y e. B C <-> A. y e. B z e. C ) )
4	2, 3	ax-mp 5	. . . . 5 |- ( z e. |^|_ y e. B C <-> A. y e. B z e. C )
5	4	rexbii 3041	. . . 4 |- ( E. x e. A z e. |^|_ y e. B C <-> E. x e. A A. y e. B z e. C )
6		eliun 4524	. . . . 5 |- ( z e. U_ x e. A C <-> E. x e. A z e. C )
7	6	ralbii 2980	. . . 4 |- ( A. y e. B z e. U_ x e. A C <-> A. y e. B E. x e. A z e. C )
8	1, 5, 7	3imtr4i 281	. . 3 |- ( E. x e. A z e. |^|_ y e. B C -> A. y e. B z e. U_ x e. A C )
9		eliun 4524	. . 3 |- ( z e. U_ x e. A |^|_ y e. B C <-> E. x e. A z e. |^|_ y e. B C )
10		eliin 4525	. . . 4 |- ( z e. _V -> ( z e. |^|_ y e. B U_ x e. A C <-> A. y e. B z e. U_ x e. A C ) )
11	2, 10	ax-mp 5	. . 3 |- ( z e. |^|_ y e. B U_ x e. A C <-> A. y e. B z e. U_ x e. A C )
12	8, 9, 11	3imtr4i 281	. 2 |- ( z e. U_ x e. A |^|_ y e. B C -> z e. |^|_ y e. B U_ x e. A C )
13	12	ssriv 3607	1 |- U_ x e. A |^|_ y e. B C C_ |^|_ y e. B U_ x e. A C

Syntax hints:   <-> wb 196   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   U_ciun 4520   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522  df-iin 4523

This theorem is referenced by: (None)