MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjssun Unicode version

Theorem disjssun 3546
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
StepHypRef Expression
1 indi 3449 . . . . 5  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  B )  u.  ( A  i^i  C ) )
21equncomi 3355 . . . 4  |-  ( A  i^i  ( B  u.  C ) )  =  ( ( A  i^i  C )  u.  ( A  i^i  B ) )
3 uneq2 3357 . . . . 5  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( ( A  i^i  C
)  u.  (/) ) )
4 un0 3513 . . . . 5  |-  ( ( A  i^i  C )  u.  (/) )  =  ( A  i^i  C )
53, 4syl6eq 2364 . . . 4  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  C )  u.  ( A  i^i  B ) )  =  ( A  i^i  C ) )
62, 5syl5eq 2360 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( A  i^i  ( B  u.  C ) )  =  ( A  i^i  C
) )
76eqeq1d 2324 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( A  i^i  ( B  u.  C ) )  =  A  <->  ( A  i^i  C )  =  A ) )
8 df-ss 3200 . 2  |-  ( A 
C_  ( B  u.  C )  <->  ( A  i^i  ( B  u.  C
) )  =  A )
9 df-ss 3200 . 2  |-  ( A 
C_  C  <->  ( A  i^i  C )  =  A )
107, 8, 93bitr4g 279 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
C_  ( B  u.  C )  <->  A  C_  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    u. cun 3184    i^i cin 3185    C_ wss 3186   (/)c0 3489
This theorem is referenced by:  hashbclem  11437  alexsubALTlem2  17794  iccntr  18378  reconnlem1  18383  dvne0  19411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490
  Copyright terms: Public domain W3C validator