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Theorem disj4 3503
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 3499 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eqcom 2285 . 2  |-  ( A  =  ( A  \  B )  <->  ( A  \  B )  =  A )
3 difss 3303 . . . 4  |-  ( A 
\  B )  C_  A
4 dfpss2 3261 . . . 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 1623    \ cdif 3149    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455
This theorem is referenced by:  marypha1lem  7186  infeq5i  7337  infpss  7843  ominf4  7938  wilthlem2  20307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456
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