Theorem un00 3862

Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
un00

Proof of Theorem un00

Step	Hyp	Ref	Expression
1		uneq12 3652	. . 3
2		un0 3810	. . 3
3	1, 2	syl6eq 2514	. 2
4		ssun1 3666	. . . . 5
5		sseq2 3525	. . . . 5
6	4, 5	mpbii 211	. . . 4
7		ss0b 3815	. . . 4
8	6, 7	sylib 196	. . 3
9		ssun2 3667	. . . . 5
10		sseq2 3525	. . . . 5
11	9, 10	mpbii 211	. . . 4
12		ss0b 3815	. . . 4
13	11, 12	sylib 196	. . 3
14	8, 13	jca 532	. 2
15	3, 14	impbii 188	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  u.cun 3473  C_wss 3475  c0 3784

This theorem is referenced by:  undisj1  3878  undisj2  3879  disjpr2  4092  rankxplim3  8320  ssxr  9675  rpnnen2  13959  wwlknext  24724  asindmre  30102

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-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785