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

Step	Hyp	Ref	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 ) )
3	1, 2	sylnibr 635	1 |- ( E. x ( x e. A /\ -. x e. B ) -> -. A C_ B )

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)