Theorem nssinpss 3856

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

Proof of Theorem nssinpss

Step	Hyp	Ref	Expression
1		inss1 3833	. . 3 |- ( A i^i B ) C_ A
2	1	biantrur 527	. 2 |- ( ( A i^i B ) =/= A <-> ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) )
3		df-ss 3588	. . 3 |- ( A C_ B <-> ( A i^i B ) = A )
4	3	necon3bbii 2841	. 2 |- ( -. A C_ B <-> ( A i^i B ) =/= A )
5		df-pss 3590	. 2 |- ( ( A i^i B ) C. A <-> ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) )
6	2, 4, 5	3bitr4i 292	1 |- ( -. A C_ B <-> ( A i^i B ) C. A )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   =/= wne 2794   i^i cin 3573   C_ wss 3574   C. wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  fbfinnfr  21645  chrelat2i  29224