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Theorem nssr 3031
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 1554 . 2  |-  ( E. x ( x  e.  A  /\  -.  x  e.  B )  ->  -.  A. x ( x  e.  A  ->  x  e.  B ) )
2 dfss2 2962 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
31, 2sylnibr 612 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 101   A.wal 1257   E.wex 1397    e. wcel 1409    C_ wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959
This theorem is referenced by: (None)
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