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Theorem nssr 3073
Description: Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
Assertion
Ref Expression
nssr  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  ->  -.  A  C_  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem nssr
StepHypRef Expression
1 exanaliim 1581 . 2  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  ->  -.  A. x ( x  e.  A  ->  x  e.  B ) )
2 dfss2 3003 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
31, 2sylnibr 635 1  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  ->  -.  A  C_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1285   E.wex 1424    e. wcel 1436    C_ wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-in 2994  df-ss 3001
This theorem is referenced by: (None)
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