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Theorem nssssr 3985
Description: Negation of subclass relationship. Compare nssr 3058. (Contributed by Jim Kingdon, 17-Sep-2018.)
Assertion
Ref Expression
nssssr  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  ->  -.  A  C_  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem nssssr
StepHypRef Expression
1 exanaliim 1579 . 2  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  ->  -.  A. x ( x  C_  A  ->  x  C_  B
) )
2 ssextss 3983 . 2  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
31, 2sylnibr 635 1  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  ->  -.  A  C_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422    C_ wss 2974
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412
This theorem is referenced by: (None)
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