Theorem disjpss 3877

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

Proof of Theorem disjpss

Step	Hyp	Ref	Expression
1		ssid 3522	. . . . . . . 8
2	1	biantru 505	. . . . . . 7
3		ssin 3719	. . . . . . 7
4	2, 3	bitri 249	. . . . . 6
5		sseq2 3525	. . . . . 6
6	4, 5	syl5bb 257	. . . . 5
7		ss0 3816	. . . . 5
8	6, 7	syl6bi 228	. . . 4
9	8	necon3ad 2667	. . 3
10	9	imp 429	. 2
11		nsspssun 3730	. . 3
12		uncom 3647	. . . 4
13	12	psseq2i 3593	. . 3
14	11, 13	bitri 249	. 2
15	10, 14	sylib 196	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  =/=wne 2652  u.cun 3473  i^icin 3474  C_wss 3475  C.wpss 3476  c0 3784

This theorem is referenced by:  isfin1-3  8787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785