Theorem nssinpss 3169

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 3157	. . 3 |- ( A i^i B ) C_ A
2	1	biantrur 287	. 2 |- ( ( A i^i B ) =/= A <-> ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) )
3		df-ss 2931	. . 3 |- ( A C_ B <-> ( A i^i B ) = A )
4	3	necon3bbii 2242	. 2 |- ( -. A C_ B <-> ( A i^i B ) =/= A )
5		df-pss 2933	. 2 |- ( ( A i^i B ) C. A <-> ( ( A i^i B ) C_ A /\ ( A i^i B ) =/= A ) )
6	2, 4, 5	3bitr4i 201	1 |- ( -. A C_ B <-> ( A i^i B ) C. A )

Syntax hints:   -. wn 3   /\ wa 97   <-> wb 98   =/= wne 2204   i^i cin 2916   C_ wss 2917   C. wpss 2918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by: (None)