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Theorem nssssr 4207
Description: Negation of subclass relationship. Compare nssr 3207. (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 1640 . 2  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  ->  -.  A. x ( x  C_  A  ->  x  C_  B
) )
2 ssextss 4205 . 2  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
31, 2sylnibr 672 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 103   A.wal 1346   E.wex 1485    C_ wss 3121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589
This theorem is referenced by: (None)
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