Theorem nsspssun 2237

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 2190	. . 3 |- B (_ (A u. B)
2	1	biantrur 724	. 2 |- (-. (A u. B) (_ B <-> (B (_ (A u. B) /\ -. (A u. B) (_ B))
3		ssid 2076	. . . . 5 |- B (_ B
4	3	biantru 723	. . . 4 |- (A (_ B <-> (A (_ B /\ B (_ B))
5		unss 2200	. . . 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 2130	. 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 2041   (_ wss 2043   (. wpss 2044

This theorem is referenced by:  disjpss 2315

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051

Copyright terms: Public domain