Theorem nssinpss 3729

Description: Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
nssinpss

Proof of Theorem nssinpss

Step	Hyp	Ref	Expression
1		inss1 3717	. . 3
2	1	biantrur 506	. 2
3		df-ss 3489	. . 3
4	3	necon3bbii 2718	. 2
5		df-pss 3491	. 2
6	2, 4, 5	3bitr4i 277	1

Syntax hints:  -.wn 3  <->wb 184  /\wa 369  =/=wne 2652  i^icin 3474  C_wss 3475  C.wpss 3476

This theorem is referenced by:  fbfinnfr  20342  chrelat2i  27284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-in 3482  df-ss 3489  df-pss 3491