Theorem disjpss 2924

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 C. (A u. B))

Proof of Theorem disjpss

Step	Hyp	Ref	Expression
1		sseq2 2639	. . . . . 6 |- ((A i^i B) = (/) -> (B C_ (A i^i B) <-> B C_ (/)))
2		ssid 2634	. . . . . . . 8 |- B C_ B
3	2	biantru 793	. . . . . . 7 |- (B C_ A <-> (B C_ A /\ B C_ B))
4		ssin 2814	. . . . . . 7 |- ((B C_ A /\ B C_ B) <-> B C_ (A i^i B))
5	3, 4	bitri 190	. . . . . 6 |- (B C_ A <-> B C_ (A i^i B))
6	1, 5	syl5bb 591	. . . . 5 |- ((A i^i B) = (/) -> (B C_ A <-> B C_ (/)))
7		ss0 2902	. . . . 5 |- (B C_ (/) -> B = (/))
8	6, 7	syl6bi 231	. . . 4 |- ((A i^i B) = (/) -> (B C_ A -> B = (/)))
9	8	necon3ad 2037	. . 3 |- ((A i^i B) = (/) -> (B =/= (/) -> -. B C_ A))
10	9	imp 377	. 2 |- (((A i^i B) = (/) /\ B =/= (/)) -> -. B C_ A)
11		nsspssun 2826	. . 3 |- (-. B C_ A <-> A C. (B u. A))
12		uncom 2744	. . . 4 |- (B u. A) = (A u. B)
13	12	psseq2i 2700	. . 3 |- (A C. (B u. A) <-> A C. (A u. B))
14	11, 13	bitri 190	. 2 |- (-. B C_ A <-> A C. (A u. B))
15	10, 14	sylib 215	1 |- (((A i^i B) = (/) /\ B =/= (/)) -> A C. (A u. B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   =/= wne 2017   u. cun 2591   i^i cin 2592   C_ wss 2593   C. wpss 2594  (/)c0 2875

This theorem is referenced by:  infxpidmlem11 8831

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876

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