Theorem djudisj 5031

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 4749	. 2 |- U_ x e. A ( { x } X. C ) C_ ( A X. _V )
2		incom 3314	. . 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 4749	. . . 4 |- U_ y e. B ( { y } X. D ) C_ ( B X. _V )
4		incom 3314	. . . . 5 |- ( ( B X. _V ) i^i ( A X. _V ) ) = ( ( A X. _V ) i^i ( B X. _V ) )
5		xpdisj1 5028	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i ( B X. _V ) ) = (/) )
6	4, 5	syl5eq 2211	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( B X. _V ) i^i ( A X. _V ) ) = (/) )
7		ssdisj 3465	. . . 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 411	. . 3 |- ( ( A i^i B ) = (/) -> ( U_ y e. B ( { y } X. D ) i^i ( A X. _V ) ) = (/) )
9	2, 8	syl5eq 2211	. 2 |- ( ( A i^i B ) = (/) -> ( ( A X. _V ) i^i U_ y e. B ( { y } X. D ) ) = (/) )
10		ssdisj 3465	. 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 411	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 1343   _Vcvv 2726   i^i cin 3115   C_ wss 3116   (/)c0 3409   {csn 3576   U_ciun 3866   X. cxp 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-iun 3868  df-opab 4044  df-xp 4610  df-rel 4611

This theorem is referenced by: (None)