Theorem nssinpss 2824

Description: Negation of subclass expressed in terms of intersection and proper subclass. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem nssinpss

Step	Hyp	Ref	Expression
1		inss1 2812	. . 3 |- (A i^i B) C_ A
2	1	biantrur 794	. 2 |- ((A i^i B) =/= A <-> ((A i^i B) C_ A /\ (A i^i B) =/= A))
3		df-ss 2605	. . 3 |- (A C_ B <-> (A i^i B) = A)
4	3	necon3bbii 2031	. 2 |- (-. A C_ B <-> (A i^i B) =/= A)
5		df-pss 2607	. 2 |- ((A i^i B) C. A <-> ((A i^i B) C_ A /\ (A i^i B) =/= A))
6	2, 4, 5	3bitr4i 200	1 |- (-. A C_ B <-> (A i^i B) C. A)

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   =/= wne 2017   i^i cin 2592   C_ wss 2593   C. wpss 2594

This theorem is referenced by:  chrelat2i 11937

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-in 2603  df-ss 2605  df-pss 2607

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