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Theorem un00 3460
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 3276 . . 3  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  ( (/)  u.  (/) ) )
2 un0 3447 . . 3  |-  ( (/)  u.  (/) )  =  (/)
31, 2eqtrdi 2219 . 2  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  u.  B )  =  (/) )
4 ssun1 3290 . . . . 5  |-  A  C_  ( A  u.  B
)
5 sseq2 3171 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( A 
C_  ( A  u.  B )  <->  A  C_  (/) ) )
64, 5mpbii 147 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  A  C_  (/) )
7 ss0b 3453 . . . 4  |-  ( A 
C_  (/)  <->  A  =  (/) )
86, 7sylib 121 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  A  =  (/) )
9 ssun2 3291 . . . . 5  |-  B  C_  ( A  u.  B
)
10 sseq2 3171 . . . . 5  |-  ( ( A  u.  B )  =  (/)  ->  ( B 
C_  ( A  u.  B )  <->  B  C_  (/) ) )
119, 10mpbii 147 . . . 4  |-  ( ( A  u.  B )  =  (/)  ->  B  C_  (/) )
12 ss0b 3453 . . . 4  |-  ( B 
C_  (/)  <->  B  =  (/) )
1311, 12sylib 121 . . 3  |-  ( ( A  u.  B )  =  (/)  ->  B  =  (/) )
148, 13jca 304 . 2  |-  ( ( A  u.  B )  =  (/)  ->  ( A  =  (/)  /\  B  =  (/) ) )
153, 14impbii 125 1  |-  ( ( A  =  (/)  /\  B  =  (/) )  <->  ( A  u.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1348    u. cun 3119    C_ wss 3121   (/)c0 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415
This theorem is referenced by:  undisj1  3471  undisj2  3472  disjpr2  3645
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