Theorem nssinpss 2240

Description: Negation of subclass expressed in terms of intersection and proper subclass.

Assertion

Ref	Expression
nssinpss	|- (-. A (_ B <-> (A i^i B) (. A)

Proof of Theorem nssinpss

Step	Hyp	Ref	Expression
1		inss1 2230	. . 3 |- (A i^i B) (_ A
2	1	biantrur 725	. 2 |- (-. A (_ (A i^i B) <-> ((A i^i B) (_ A /\ -. A (_ (A i^i B)))
3		ssid 2080	. . . . 5 |- A (_ A
4	3	biantrur 725	. . . 4 |- (A (_ B <-> (A (_ A /\ A (_ B))
5		ssin 2232	. . . 4 |- ((A (_ A /\ A (_ B) <-> A (_ (A i^i B))
6	4, 5	bitr 173	. . 3 |- (A (_ B <-> A (_ (A i^i B))
7	6	negbii 187	. 2 |- (-. A (_ B <-> -. A (_ (A i^i B))
8		dfpss3 2134	. 2 |- ((A i^i B) (. A <-> ((A i^i B) (_ A /\ -. A (_ (A i^i B)))
9	2, 7, 8	3bitr4 183	1 |- (-. A (_ B <-> (A i^i B) (. A)

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   i^i cin 2046   (_ wss 2047   (. wpss 2048

This theorem is referenced by:  chrelat2 10292

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-in 2051  df-ss 2053  df-pss 2055

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