Theorem disjpss 2323

Description: A class is a proper subset of its union with a disjoint nonempty class.

Assertion

Ref	Expression
disjpss	|- (((A i^i B) = (/) /\ B =/= (/)) -> A (. (A u. B))

Proof of Theorem disjpss

Step	Hyp	Ref	Expression
1		sseq2 2086	. . . . . 6 |- ((A i^i B) = (/) -> (B (_ (A i^i B) <-> B (_ (/)))
2		ssid 2083	. . . . . . . 8 |- B (_ B
3	2	biantru 726	. . . . . . 7 |- (B (_ A <-> (B (_ A /\ B (_ B))
4		ssin 2235	. . . . . . 7 |- ((B (_ A /\ B (_ B) <-> B (_ (A i^i B))
5	3, 4	bitr 173	. . . . . 6 |- (B (_ A <-> B (_ (A i^i B))
6	1, 5	syl5bb 534	. . . . 5 |- ((A i^i B) = (/) -> (B (_ A <-> B (_ (/)))
7		ss0 2307	. . . . 5 |- (B (_ (/) -> B = (/))
8	6, 7	syl6bi 214	. . . 4 |- ((A i^i B) = (/) -> (B (_ A -> B = (/)))
9	8	necon3ad 1605	. . 3 |- ((A i^i B) = (/) -> (B =/= (/) -> -. B (_ A))
10	9	imp 350	. 2 |- (((A i^i B) = (/) /\ B =/= (/)) -> -. B (_ A)
11		nsspssun 2244	. . 3 |- (-. B (_ A <-> A (. (B u. A))
12		uncom 2179	. . . 4 |- (B u. A) = (A u. B)
13	12	psseq2i 2141	. . 3 |- (A (. (B u. A) <-> A (. (A u. B))
14	11, 13	bitr 173	. 2 |- (-. B (_ A <-> A (. (A u. B))
15	10, 14	sylib 198	1 |- (((A i^i B) = (/) /\ B =/= (/)) -> A (. (A u. B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 958   =/= wne 1588   u. cun 2048   i^i cin 2049   (_ wss 2050   (. wpss 2051  (/)c0 2283

This theorem is referenced by:  infxpidmlem11 7563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284

Copyright terms: Public domain