Theorem un00 4011

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

Assertion

Ref	Expression
un00	|- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )

Proof of Theorem un00

Step	Hyp	Ref	Expression
1		uneq12 3762	. . 3 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = ( (/) u. (/) ) )
2		un0 3967	. . 3 |- ( (/) u. (/) ) = (/)
3	1, 2	syl6eq 2672	. 2 |- ( ( A = (/) /\ B = (/) ) -> ( A u. B ) = (/) )
4		ssun1 3776	. . . . 5 |- A C_ ( A u. B )
5		sseq2 3627	. . . . 5 |- ( ( A u. B ) = (/) -> ( A C_ ( A u. B ) <-> A C_ (/) ) )
6	4, 5	mpbii 223	. . . 4 |- ( ( A u. B ) = (/) -> A C_ (/) )
7		ss0b 3973	. . . 4 |- ( A C_ (/) <-> A = (/) )
8	6, 7	sylib 208	. . 3 |- ( ( A u. B ) = (/) -> A = (/) )
9		ssun2 3777	. . . . 5 |- B C_ ( A u. B )
10		sseq2 3627	. . . . 5 |- ( ( A u. B ) = (/) -> ( B C_ ( A u. B ) <-> B C_ (/) ) )
11	9, 10	mpbii 223	. . . 4 |- ( ( A u. B ) = (/) -> B C_ (/) )
12		ss0b 3973	. . . 4 |- ( B C_ (/) <-> B = (/) )
13	11, 12	sylib 208	. . 3 |- ( ( A u. B ) = (/) -> B = (/) )
14	8, 13	jca 554	. 2 |- ( ( A u. B ) = (/) -> ( A = (/) /\ B = (/) ) )
15	3, 14	impbii 199	1 |- ( ( A = (/) /\ B = (/) ) <-> ( A u. B ) = (/) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   u. cun 3572   C_ wss 3574   (/)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-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  undisj1  4029  undisj2  4030  disjpr2  4248  disjpr2OLD  4249  rankxplim3  8744  ssxr  10107  rpnnen2lem12  14954  wwlksnext  26788  asindmre  33495  iunrelexp0  37994  uneqsn  38321