Theorem nsspssun 3489

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 3428	. . . 4 |- B C_ (A u. B)
2	1	biantrur 492	. . 3 |- (-. (A u. B) C_ B <-> (B C_ (A u. B) /\ -. (A u. B) C_ B))
3		ssid 3291	. . . . 5 |- B C_ B
4	3	biantru 491	. . . 4 |- (A C_ B <-> (A C_ B /\ B C_ B))
5		unss 3438	. . . 4 |- ((A C_ B /\ B C_ B) <-> (A u. B) C_ B)
6	4, 5	bitri 240	. . 3 |- (A C_ B <-> (A u. B) C_ B)
7	2, 6	xchnxbir 300	. 2 |- (-. A C_ B <-> (B C_ (A u. B) /\ -. (A u. B) C_ B))
8		dfpss3 3356	. 2 |- (B C. (A u. B) <-> (B C_ (A u. B) /\ -. (A u. B) C_ B))
9	7, 8	bitr4i 243	1 |- (-. A C_ B <-> B C. (A u. B))

Syntax hints:  -. wn 3   <-> wb 176   /\ wa 358   u. cun 3208   C_ wss 3258   C. wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260  df-pss 3262

This theorem is referenced by:  disjpss  3602