Theorem disjssun 4036

Description: Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
disjssun	|- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) )

Proof of Theorem disjssun

Step	Hyp	Ref	Expression
1		uneq2 3761	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( ( A i^i C ) u. (/) ) )
2		indi 3873	. . . . 5 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
3	2	equncomi 3759	. . . 4 |- ( A i^i ( B u. C ) ) = ( ( A i^i C ) u. ( A i^i B ) )
4		un0 3967	. . . . 5 |- ( ( A i^i C ) u. (/) ) = ( A i^i C )
5	4	eqcomi 2631	. . . 4 |- ( A i^i C ) = ( ( A i^i C ) u. (/) )
6	1, 3, 5	3eqtr4g 2681	. . 3 |- ( ( A i^i B ) = (/) -> ( A i^i ( B u. C ) ) = ( A i^i C ) )
7	6	eqeq1d 2624	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i ( B u. C ) ) = A <-> ( A i^i C ) = A ) )
8		df-ss 3588	. 2 |- ( A C_ ( B u. C ) <-> ( A i^i ( B u. C ) ) = A )
9		df-ss 3588	. 2 |- ( A C_ C <-> ( A i^i C ) = A )
10	7, 8, 9	3bitr4g 303	1 |- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915

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-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  hashbclem  13236  alexsubALTlem2  21852  iccntr  22624  reconnlem1  22629  dvne0  23774