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Theorem nssssr 3958
Description: Negation of subclass relationship. Compare nssr 3003. (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 1538 . 2  |-  ( E. x ( x  C_  A  /\  -.  x  C_  B )  ->  -.  A. x ( x  C_  A  ->  x  C_  B
) )
2 ssextss 3956 . 2  |-  ( A 
C_  B  <->  A. x
( x  C_  A  ->  x  C_  B )
)
31, 2sylnibr 602 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 97   A.wal 1241   E.wex 1381    C_ wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381
This theorem is referenced by: (None)
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