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Theorem dfss6 36375
Description: Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)
Assertion
Ref Expression
dfss6  |-  ( A 
C_  B  <->  -.  E. x
( x  e.  A  /\  -.  x  e.  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem dfss6
StepHypRef Expression
1 dfss2 3423 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
2 notnot 293 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  -.  -.  A. x ( x  e.  A  ->  x  e.  B ) )
31, 2bitri 253 . 2  |-  ( A 
C_  B  <->  -.  -.  A. x ( x  e.  A  ->  x  e.  B ) )
4 exanali 1723 . 2  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A. x
( x  e.  A  ->  x  e.  B ) )
53, 4xchbinxr 313 1  |-  ( A 
C_  B  <->  -.  E. x
( x  e.  A  /\  -.  x  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1444   E.wex 1665    e. wcel 1889    C_ wss 3406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3413  df-ss 3420
This theorem is referenced by: (None)
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