Theorem djudisj 5181

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 4891	. 2 |- U_ x e. A ( { x } X. C ) C_ ( A X. _V )
2		incom 3410	. . 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 4891	. . . 4 |- U_ y e. B ( { y } X. D ) C_ ( B X. _V )
4		incom 3410	. . . . 5 |- ( ( B X. _V ) i^i ( A X. _V ) ) = ( ( A X. _V ) i^i ( B X. _V ) )
5		xpdisj1 5178	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i ( B X. _V ) ) = (/) )
6	4, 5	eqtrid 2277	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( B X. _V ) i^i ( A X. _V ) ) = (/) )
7		ssdisj 3562	. . . 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 2277	. 2 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i U_ y e. B ( { y } X. D ) ) = (/) )
10		ssdisj 3562	. 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 1398   _Vcvv 2812   i^i cin 3209   C_ wss 3210   (/)c0 3505   {csn 3682   U_ciun 3984   X. cxp 4738

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-iun 3986  df-opab 4165  df-xp 4746  df-rel 4747

This theorem is referenced by: (None)