Theorem nsspssun 3857

Description: Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
nsspssun	|- ( -. A C_ B <-> B C. ( A u. B ) )

Proof of Theorem nsspssun

Step	Hyp	Ref	Expression
1		ssun2 3777	. . . 4 |- B C_ ( A u. B )
2	1	biantrur 527	. . 3 |- ( -. ( A u. B ) C_ B <-> ( B C_ ( A u. B ) /\ -. ( A u. B ) C_ B ) )
3		ssid 3624	. . . . 5 |- B C_ B
4	3	biantru 526	. . . 4 |- ( A C_ B <-> ( A C_ B /\ B C_ B ) )
5		unss 3787	. . . 4 |- ( ( A C_ B /\ B C_ B ) <-> ( A u. B ) C_ B )
6	4, 5	bitri 264	. . 3 |- ( A C_ B <-> ( A u. B ) C_ B )
7	2, 6	xchnxbir 323	. 2 |- ( -. A C_ B <-> ( B C_ ( A u. B ) /\ -. ( A u. B ) C_ B ) )
8		dfpss3 3693	. 2 |- ( B C. ( A u. B ) <-> ( B C_ ( A u. B ) /\ -. ( A u. B ) C_ B ) )
9	7, 8	bitr4i 267	1 |- ( -. A C_ B <-> B C. ( A u. B ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   u. cun 3572   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-un 3579  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  disjpss  4028  lindsenlbs  33404