Theorem nss 2109

Description: Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18.

Assertion

Ref	Expression
nss	|- (-. A (_ B <-> E.x(x e. A /\ -. x e. B))

Distinct variable groups:   x,A   x,B

Proof of Theorem nss

Step	Hyp	Ref	Expression
1		exnal 1036	. 2 |- (E.x -. (x e. A -> x e. B) <-> -. A.x(x e. A -> x e. B))
2		annim 238	. . 3 |- ((x e. A /\ -. x e. B) <-> -. (x e. A -> x e. B))
3	2	exbii 1049	. 2 |- (E.x(x e. A /\ -. x e. B) <-> E.x -. (x e. A -> x e. B))
4		dfss2 2054	. . 3 |- (A (_ B <-> A.x(x e. A -> x e. B))
5	4	negbii 187	. 2 |- (-. A (_ B <-> -. A.x(x e. A -> x e. B))
6	1, 3, 5	3bitr4r 184	1 |- (-. A (_ B <-> E.x(x e. A /\ -. x e. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   e. wcel 956  E.wex 978   (_ wss 2043

This theorem is referenced by:  psslinpr 5115  reclem2pr 5137  shne0 9309

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-in 2047  df-ss 2049

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