Theorem nsspssun 2241

Description: Negation of subclass expressed in terms of proper subclass and union.

Assertion

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

Proof of Theorem nsspssun

Step	Hyp	Ref	Expression
1		ssun2 2194	. . 3 |- B (_ (A u. B)
2	1	biantrur 725	. 2 |- (-. (A u. B) (_ B <-> (B (_ (A u. B) /\ -. (A u. B) (_ B))
3		ssid 2080	. . . . 5 |- B (_ B
4	3	biantru 724	. . . 4 |- (A (_ B <-> (A (_ B /\ B (_ B))
5		unss 2204	. . . 4 |- ((A (_ B /\ B (_ B) <-> (A u. B) (_ B)
6	4, 5	bitr 173	. . 3 |- (A (_ B <-> (A u. B) (_ B)
7	6	negbii 187	. 2 |- (-. A (_ B <-> -. (A u. B) (_ B)
8		dfpss3 2134	. 2 |- (B (. (A u. B) <-> (B (_ (A u. B) /\ -. (A u. B) (_ B))
9	2, 7, 8	3bitr4 183	1 |- (-. A (_ B <-> B (. (A u. B))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   u. cun 2045   (_ wss 2047   (. wpss 2048

This theorem is referenced by:  disjpss 2319

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055

Copyright terms: Public domain