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Theorem disj4 3841
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 3837 . 2  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eqcom 2431 . 2  |-  ( A  =  ( A  \  B )  <->  ( A  \  B )  =  A )
3 difss 3592 . . . 4  |-  ( A 
\  B )  C_  A
4 dfpss2 3550 . . . 4  |-  ( ( A  \  B ) 
C.  A  <->  ( ( A  \  B )  C_  A  /\  -.  ( A 
\  B )  =  A ) )
53, 4mpbiran 926 . . 3  |-  ( ( A  \  B ) 
C.  A  <->  -.  ( A  \  B )  =  A )
65con2bii 333 . 2  |-  ( ( A  \  B )  =  A  <->  -.  ( A  \  B )  C.  A )
71, 2, 63bitri 274 1  |-  ( ( A  i^i  B )  =  (/)  <->  -.  ( A  \  B )  C.  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    = wceq 1437    \ cdif 3433    i^i cin 3435    C_ wss 3436    C. wpss 3437   (/)c0 3761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-v 3083  df-dif 3439  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762
This theorem is referenced by:  marypha1lem  7950  infeq5i  8144  wilthlem2  23981  topdifinffinlem  31691
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