Theorem djudisj 5058

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 4774	. 2 |- U_ x e. A ( { x } X. C ) C_ ( A X. _V )
2		incom 3329	. . 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 4774	. . . 4 |- U_ y e. B ( { y } X. D ) C_ ( B X. _V )
4		incom 3329	. . . . 5 |- ( ( B X. _V ) i^i ( A X. _V ) ) = ( ( A X. _V ) i^i ( B X. _V ) )
5		xpdisj1 5055	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i ( B X. _V ) ) = (/) )
6	4, 5	eqtrid 2222	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( B X. _V ) i^i ( A X. _V ) ) = (/) )
7		ssdisj 3481	. . . 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 414	. . 3 |- ( ( A i^i B ) = (/) -> ( U_ y e. B ( { y } X. D ) i^i ( A X. _V ) ) = (/) )
9	2, 8	eqtrid 2222	. 2 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i U_ y e. B ( { y } X. D ) ) = (/) )
10		ssdisj 3481	. 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 414	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 1353   _Vcvv 2739   i^i cin 3130   C_ wss 3131   (/)c0 3424   {csn 3594   U_ciun 3888   X. cxp 4626

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-iun 3890  df-opab 4067  df-xp 4634  df-rel 4635

This theorem is referenced by: (None)