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Theorem dfss6 36221
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 3453 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
2 notnot 292 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  -.  -.  A. x ( x  e.  A  ->  x  e.  B ) )
31, 2bitri 252 . 2  |-  ( A 
C_  B  <->  -.  -.  A. x ( x  e.  A  ->  x  e.  B ) )
4 exanali 1715 . 2  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  <->  -.  A. x
( x  e.  A  ->  x  e.  B ) )
53, 4xchbinxr 312 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 187    /\ wa 370   A.wal 1435   E.wex 1659    e. wcel 1868    C_ wss 3436
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-in 3443  df-ss 3450
This theorem is referenced by: (None)
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