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Theorem disj4 3676
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 3672 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eqcom 2438 . 2  |-  ( A  =  ( A  \  B )  <->  ( A  \  B )  =  A )
3 difss 3474 . . . 4  |-  ( A 
\  B )  C_  A
4 dfpss2 3432 . . . 4  |-  ( ( A  \  B ) 
C.  A  <->  ( ( A  \  B )  C_  A  /\  -.  ( A 
\  B )  =  A ) )
53, 4mpbiran 885 . . 3  |-  ( ( A  \  B ) 
C.  A  <->  -.  ( A  \  B )  =  A )
65con2bii 323 . 2  |-  ( ( A  \  B )  =  A  <->  -.  ( A  \  B )  C.  A )
71, 2, 63bitri 263 1  |-  ( ( A  i^i  B )  =  (/)  <->  -.  ( A  \  B )  C.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    \ cdif 3317    i^i cin 3319    C_ wss 3320    C. wpss 3321   (/)c0 3628
This theorem is referenced by:  marypha1lem  7438  infeq5i  7591  wilthlem2  20852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629
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