Theorem djudisj 5561

Description: Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Assertion

Ref	Expression
djudisj	|- ( ( A i^i B ) = (/) -> ( U_ x e. A ( { x } X. C ) i^i U_ y e. B ( { y } X. D ) ) = (/) )

Distinct variable groups:   x, A   y, B

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

Proof of Theorem djudisj

Step	Hyp	Ref	Expression
1		djussxp 5267	. 2 |- U_ x e. A ( { x } X. C ) C_ ( A X. _V )
2		incom 3805	. . 3 |- ( ( A X. _V ) i^i U_ y e. B ( { y } X. D ) ) = ( U_ y e. B ( { y } X. D ) i^i ( A X. _V ) )
3		djussxp 5267	. . . 4 |- U_ y e. B ( { y } X. D ) C_ ( B X. _V )
4		incom 3805	. . . . 5 |- ( ( B X. _V ) i^i ( A X. _V ) ) = ( ( A X. _V ) i^i ( B X. _V ) )
5		xpdisj1 5555	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i ( B X. _V ) ) = (/) )
6	4, 5	syl5eq 2668	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( B X. _V ) i^i ( A X. _V ) ) = (/) )
7		ssdisj 4026	. . . 4 |- ( ( U_ y e. B ( { y } X. D ) C_ ( B X. _V ) /\ ( ( B X. _V ) i^i ( A X. _V ) ) = (/) ) -> ( U_ y e. B ( { y } X. D ) i^i ( A X. _V ) ) = (/) )
8	3, 6, 7	sylancr 695	. . 3 |- ( ( A i^i B ) = (/) -> ( U_ y e. B ( { y } X. D ) i^i ( A X. _V ) ) = (/) )
9	2, 8	syl5eq 2668	. 2 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i U_ y e. B ( { y } X. D ) ) = (/) )
10		ssdisj 4026	. 2 |- ( ( U_ x e. A ( { x } X. C ) C_ ( A X. _V ) /\ ( ( A X. _V ) i^i U_ y e. B ( { y } X. D ) ) = (/) ) -> ( U_ x e. A ( { x } X. C ) i^i U_ y e. B ( { y } X. D ) ) = (/) )
11	1, 9, 10	sylancr 695	1 |- ( ( A i^i B ) = (/) -> ( U_ x e. A ( { x } X. C ) i^i U_ y e. B ( { y } X. D ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   X. cxp 5112

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  ackbij1lem9  9050