Theorem nssinpss 1665

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 1657	. . 3 |- (A i^i B) (_ A
2	1	biantrur 544	. 2 |- (-. A (_ (A i^i B) <-> ((A i^i B) (_ A /\ -. A (_ (A i^i B)))
3		ssid 1519	. . . . 5 |- A (_ A
4	3	biantrur 544	. . . 4 |- (A (_ B <-> (A (_ A /\ A (_ B))
5		ssin 1659	. . . 4 |- ((A (_ A /\ A (_ B) <-> A (_ (A i^i B))
6	4, 5	bitr 151	. . 3 |- (A (_ B <-> A (_ (A i^i B))
7	6	negbii 162	. 2 |- (-. A (_ B <-> -. A (_ (A i^i B))
8		dfpss3 1558	. 2 |- ((A i^i B) (. A <-> ((A i^i B) (_ A /\ -. A (_ (A i^i B)))
9	2, 7, 8	3bitr4 158	1 |- (-. A (_ B <-> (A i^i B) (. A)

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196   i^i cin 1486   (_ wss 1487   (. wpss 1488

This theorem is referenced by:  chrelat2 5758

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-v 1349  df-in 1491  df-ss 1492  df-pss 1494

metamath.org