Theorem disjpss 4028

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 3624	. . . . . . . 8 |- B C_ B
2	1	biantru 526	. . . . . . 7 |- ( B C_ A <-> ( B C_ A /\ B C_ B ) )
3		ssin 3835	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) <-> B C_ ( A i^i B ) )
4	2, 3	bitri 264	. . . . . 6 |- ( B C_ A <-> B C_ ( A i^i B ) )
5		sseq2 3627	. . . . . 6 |- ( ( A i^i B ) = (/) -> ( B C_ ( A i^i B ) <-> B C_ (/) ) )
6	4, 5	syl5bb 272	. . . . 5 |- ( ( A i^i B ) = (/) -> ( B C_ A <-> B C_ (/) ) )
7		ss0 3974	. . . . 5 |- ( B C_ (/) -> B = (/) )
8	6, 7	syl6bi 243	. . . 4 |- ( ( A i^i B ) = (/) -> ( B C_ A -> B = (/) ) )
9	8	necon3ad 2807	. . 3 |- ( ( A i^i B ) = (/) -> ( B =/= (/) -> -. B C_ A ) )
10	9	imp 445	. 2 |- ( ( ( A i^i B ) = (/) /\ B =/= (/) ) -> -. B C_ A )
11		nsspssun 3857	. . 3 |- ( -. B C_ A <-> A C. ( B u. A ) )
12		uncom 3757	. . . 4 |- ( B u. A ) = ( A u. B )
13	12	psseq2i 3697	. . 3 |- ( A C. ( B u. A ) <-> A C. ( A u. B ) )
14	11, 13	bitri 264	. 2 |- ( -. B C_ A <-> A C. ( A u. B ) )
15	10, 14	sylib 208	1 |- ( ( ( A i^i B ) = (/) /\ B =/= (/) ) -> A C. ( A u. B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   u. cun 3572   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916

This theorem is referenced by:  isfin1-3  9208