MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  un00 Unicode version

Theorem un00 3490
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
StepHypRef Expression
1 uneq12 3324 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
2 un0 3479 . . 3  |-  ( (/)  u.  (/) )  =  (/)
31, 2syl6eq 2331 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  (/) )
4 ssun1 3338 . . . . 5  |-  A  C_  ( A  u.  B
)
5 sseq2 3200 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( A 
C_  ( A  u.  B )  <->  A  C_  (/) ) )
64, 5mpbii 202 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  A  C_  (/) )
7 ss0b 3484 . . . 4  |-  ( A 
C_  (/)  <->  A  =  (/) )
86, 7sylib 188 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  A  =  (/) )
9 ssun2 3339 . . . . 5  |-  B  C_  ( A  u.  B
)
10 sseq2 3200 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( B 
C_  ( A  u.  B )  <->  B  C_  (/) ) )
119, 10mpbii 202 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  B  C_  (/) )
12 ss0b 3484 . . . 4  |-  ( B 
C_  (/)  <->  B  =  (/) )
1311, 12sylib 188 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  B  =  (/) )
148, 13jca 518 . 2  |-  ( ( A  u.  B )  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) )
153, 14impbii 180 1  |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    u. cun 3150    C_ wss 3152   (/)c0 3455
This theorem is referenced by:  undisj1  3506  undisj2  3507  rankxplim3  7551  ssxr  8892  rpnnen2  12504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456
  Copyright terms: Public domain W3C validator