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Theorem disj4 3537
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.)
Assertion
Ref Expression
disj4  |-  ( ( A  i^i  B )  =  (/)  <->  -.  ( A  \  B )  C.  A
)

Proof of Theorem disj4
StepHypRef Expression
1 disj3 3533 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eqcom 2318 . 2  |-  ( A  =  ( A  \  B )  <->  ( A  \  B )  =  A )
3 difss 3337 . . . 4  |-  ( A 
\  B )  C_  A
4 dfpss2 3295 . . . 4  |-  ( ( A  \  B ) 
C.  A  <->  ( ( A  \  B )  C_  A  /\  -.  ( A 
\  B )  =  A ) )
53, 4mpbiran 884 . . 3  |-  ( ( A  \  B ) 
C.  A  <->  -.  ( A  \  B )  =  A )
65con2bii 322 . 2  |-  ( ( A  \  B )  =  A  <->  -.  ( A  \  B )  C.  A )
71, 2, 63bitri 262 1  |-  ( ( A  i^i  B )  =  (/)  <->  -.  ( A  \  B )  C.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1633    \ cdif 3183    i^i cin 3185    C_ wss 3186    C. wpss 3187   (/)c0 3489
This theorem is referenced by:  marypha1lem  7231  infeq5i  7382  infpss  7888  ominf4  7983  wilthlem2  20360
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-ne 2481  df-ral 2582  df-v 2824  df-dif 3189  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490
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