Theorem disjpss 3278

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 2964	. . . . . . . 8 |- B C_ B
2	1	biantru 286	. . . . . . 7 |- ( B C_ A <-> ( B C_ A /\ B C_ B ) )
3		ssin 3159	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) <-> B C_ ( A i^i B ) )
4	2, 3	bitri 173	. . . . . 6 |- ( B C_ A <-> B C_ ( A i^i B ) )
5		sseq2 2967	. . . . . 6 |- ( ( A i^i B ) = (/) -> ( B C_ ( A i^i B ) <-> B C_ (/) ) )
6	4, 5	syl5bb 181	. . . . 5 |- ( ( A i^i B ) = (/) -> ( B C_ A <-> B C_ (/) ) )
7		ss0 3257	. . . . 5 |- ( B C_ (/) -> B = (/) )
8	6, 7	syl6bi 152	. . . 4 |- ( ( A i^i B ) = (/) -> ( B C_ A -> B = (/) ) )
9	8	necon3ad 2247	. . 3 |- ( ( A i^i B ) = (/) -> ( B =/= (/) -> -. B C_ A ) )
10	9	imp 115	. 2 |- ( ( ( A i^i B ) = (/) /\ B =/= (/) ) -> -. B C_ A )
11		nsspssun 3170	. . 3 |- ( -. B C_ A <-> A C. ( B u. A ) )
12		uncom 3087	. . . 4 |- ( B u. A ) = ( A u. B )
13	12	psseq2i 3034	. . 3 |- ( A C. ( B u. A ) <-> A C. ( A u. B ) )
14	11, 13	bitri 173	. 2 |- ( -. B C_ A <-> A C. ( A u. B ) )
15	10, 14	sylib 127	1 |- ( ( ( A i^i B ) = (/) /\ B =/= (/) ) -> A C. ( A u. B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   =/= wne 2204   u. cun 2915   i^i cin 2916   C_ wss 2917   C. wpss 2918   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pss 2933  df-nul 3225

This theorem is referenced by: (None)