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Theorem ss0b 3426
Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
ss0b  |-  ( A 
C_  (/)  <->  A  =  (/) )

Proof of Theorem ss0b
StepHypRef Expression
1 0ss 3425 . . 3  |-  (/)  C_  A
2 eqss 3136 . . 3  |-  ( A  =  (/)  <->  ( A  C_  (/) 
/\  (/)  C_  A )
)
31, 2mpbiran2 890 . 2  |-  ( A  =  (/)  <->  A  C_  (/) )
43bicomi 195 1  |-  ( A 
C_  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    C_ wss 3094   (/)c0 3397
This theorem is referenced by:  ss0  3427  un00  3432  ssdisj  3446  pw0  3703  fnsuppeq0  5632  cnfcom2lem  7337  card0  7524  kmlem5  7713  cf0  7810  fin1a2lem12  7970  mreexexlem3d  13475  efgval  14953  ppttop  16671  0nnei  16776  sssu  24473  filnetlem4  25662  pnonsingN  29252  osumcllem4N  29278
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-dif 3097  df-in 3101  df-ss 3108  df-nul 3398
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