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Theorem disj4 3863
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 3859 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eqcom 2463 . 2  |-  ( A  =  ( A  \  B )  <->  ( A  \  B )  =  A )
3 difss 3617 . . . 4  |-  ( A 
\  B )  C_  A
4 dfpss2 3575 . . . 4  |-  ( ( A  \  B ) 
C.  A  <->  ( ( A  \  B )  C_  A  /\  -.  ( A 
\  B )  =  A ) )
53, 4mpbiran 916 . . 3  |-  ( ( A  \  B ) 
C.  A  <->  -.  ( A  \  B )  =  A )
65con2bii 330 . 2  |-  ( ( A  \  B )  =  A  <->  -.  ( A  \  B )  C.  A )
71, 2, 63bitri 271 1  |-  ( ( A  i^i  B )  =  (/)  <->  -.  ( A  \  B )  C.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1398    \ cdif 3458    i^i cin 3460    C_ wss 3461    C. wpss 3462   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784
This theorem is referenced by:  marypha1lem  7885  infeq5i  8044  wilthlem2  23541
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