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Theorem un00 3503
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 3337 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
2 un0 3492 . . 3  |-  ( (/)  u.  (/) )  =  (/)
31, 2syl6eq 2344 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  (/) )
4 ssun1 3351 . . . . 5  |-  A  C_  ( A  u.  B
)
5 sseq2 3213 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( A 
C_  ( A  u.  B )  <->  A  C_  (/) ) )
64, 5mpbii 202 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  A  C_  (/) )
7 ss0b 3497 . . . 4  |-  ( A 
C_  (/)  <->  A  =  (/) )
86, 7sylib 188 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  A  =  (/) )
9 ssun2 3352 . . . . 5  |-  B  C_  ( A  u.  B
)
10 sseq2 3213 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( B 
C_  ( A  u.  B )  <->  B  C_  (/) ) )
119, 10mpbii 202 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  B  C_  (/) )
12 ss0b 3497 . . . 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 1632    u. cun 3163    C_ wss 3165   (/)c0 3468
This theorem is referenced by:  undisj1  3519  undisj2  3520  rankxplim3  7567  ssxr  8908  rpnnen2  12520  disjpr2  28185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469
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