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Theorem un00 3663
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 3496 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
2 un0 3652 . . 3  |-  ( (/)  u.  (/) )  =  (/)
31, 2syl6eq 2484 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  (/) )
4 ssun1 3510 . . . . 5  |-  A  C_  ( A  u.  B
)
5 sseq2 3370 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( A 
C_  ( A  u.  B )  <->  A  C_  (/) ) )
64, 5mpbii 203 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  A  C_  (/) )
7 ss0b 3657 . . . 4  |-  ( A 
C_  (/)  <->  A  =  (/) )
86, 7sylib 189 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  A  =  (/) )
9 ssun2 3511 . . . . 5  |-  B  C_  ( A  u.  B
)
10 sseq2 3370 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( B 
C_  ( A  u.  B )  <->  B  C_  (/) ) )
119, 10mpbii 203 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  B  C_  (/) )
12 ss0b 3657 . . . 4  |-  ( B 
C_  (/)  <->  B  =  (/) )
1311, 12sylib 189 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  B  =  (/) )
148, 13jca 519 . 2  |-  ( ( A  u.  B )  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) )
153, 14impbii 181 1  |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    u. cun 3318    C_ wss 3320   (/)c0 3628
This theorem is referenced by:  undisj1  3679  undisj2  3680  disjpr2  3870  rankxplim3  7805  ssxr  9145  rpnnen2  12825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629
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