Theorem disjpss 3602

Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)

Assertion

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

Proof of Theorem disjpss

Step	Hyp	Ref	Expression
1		ssid 3291	. . . . . . . 8 |- B C_ B
2	1	biantru 491	. . . . . . 7 |- (B C_ A <-> (B C_ A /\ B C_ B))
3		ssin 3478	. . . . . . 7 |- ((B C_ A /\ B C_ B) <-> B C_ (A i^i B))
4	2, 3	bitri 240	. . . . . 6 |- (B C_ A <-> B C_ (A i^i B))
5		sseq2 3294	. . . . . 6 |- ((A i^i B) = (/) -> (B C_ (A i^i B) <-> B C_ (/)))
6	4, 5	syl5bb 248	. . . . 5 |- ((A i^i B) = (/) -> (B C_ A <-> B C_ (/)))
7		ss0 3582	. . . . 5 |- (B C_ (/) -> B = (/))
8	6, 7	syl6bi 219	. . . 4 |- ((A i^i B) = (/) -> (B C_ A -> B = (/)))
9	8	necon3ad 2553	. . 3 |- ((A i^i B) = (/) -> (B =/= (/) -> -. B C_ A))
10	9	imp 418	. 2 |- (((A i^i B) = (/) /\ B =/= (/)) -> -. B C_ A)
11		nsspssun 3489	. . 3 |- (-. B C_ A <-> A C. (B u. A))
12		uncom 3409	. . . 4 |- (B u. A) = (A u. B)
13	12	psseq2i 3360	. . 3 |- (A C. (B u. A) <-> A C. (A u. B))
14	11, 13	bitri 240	. 2 |- (-. B C_ A <-> A C. (A u. B))
15	10, 14	sylib 188	1 |- (((A i^i B) = (/) /\ B =/= (/)) -> A C. (A u. B))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   =/= wne 2517   u. cun 3208   i^i cin 3209   C_ wss 3258   C. wpss 3259  (/)c0 3551

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-dif 3216  df-ss 3260  df-pss 3262  df-nul 3552

This theorem is referenced by: (None)